Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves By Shinichi Mochizuki ∗ Abstract In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fun- damental group of the full proper curve. We then apply this theory to show that a certain absolute p-adic version of the Grothendieck Conjec- ture holds for hyperbolic curves “of Belyi type”. This includes, in particu- lar, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hy- perbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama]. Contents: §0. Notations and Conventions §1. Maximal Abelian Cuspidalizations §2. Points and Functions §3. Maximal Pro-l Cuspidalizations Appendix. Free Lie Algebras 2000 Mathematics Subject Classification(s). 14H30. ∗ Research Institute for Mathematical Sciences, Kyoto University 2 Shinichi Mochizuki Introduction Let X be a proper hyperbolic curve over a field k which is either finite or nonarchimedean local of mixed characteristic; let U ⊆ X be an open subscheme of X. Write Π X for the étale fundamental group of X. In this paper, we study the extent to which the étale fundamental group of U may be group-theoretically reconstructed from Π X . In §1, we show that the abelian portion of the extension of Π X determined by the étale fundamental group of U may be group-theoretically reconstructed from Π X [cf. Theorem 1.1, (iii)], and, moreover, that this construction has certain remarkable rigidity properties [cf. Propositions 1.10, (i); 2.3, (i)]. In §2, we show that this abelian portion of the extension is sufficient to reconstruct [in essence] the multiplicative group of the function field of X [cf. Theorem 2.1, (ii)]. In the case of nonarchimedean [mixed-characteristic] local fields, this already implies various interesting consequences in the context of the absolute anabelian geometry studied in [Mzk5], [Mzk6], [Mzk8]. In particular, it implies that the absolute p-adic version of the Grothendieck Conjecture [i.e., an absolute version of [a certain portion of] the relative result that appears as the main result of [Mzk4]] holds for hyperbolic curves “of Belyi type” [cf. Definition 2.3; Corollary 2.3]. This includes, in particular, hyperbolic curves “of strictly Belyi type”, i.e., affine hyperbolic curves over a nonarchimedean [mixed- characteristic] local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. In particular, we obtain a new countable class of “absolute curves” [in the terminology of [Mzk6]], whose absoluteness is, in certain respects, reminiscent of the absoluteness of the canonical curves of p-adic Teichmüller theory discussed in [Mzk6] [cf. Remark 30], but [in contrast to the class of canonical curves] appears [at least from the point of view of certain circumstantial evidence] unlikely to be Zariski dense in most moduli spaces [cf. Remark 31]. Finally, in §3, we apply the theory of the weight filtration [cf., e.g., [Kane], [Mtm]], together with various generalities concerning free Lie algebras [cf. the Appendix], to develop various “higher order generalizations” of the theory of §1, 2. In particular, we obtain various “higher order generalizations” of the “remarkable rigidity” referred to above [cf. Propositions 3.4, 3.6, especially Proposition 3.6, (iii)], which we apply to show that, relative to the notation introduced above, the geometrically pro-l portion [where l is a prime number invertible in k] of the étale fundamental group of U may be recovered from Π X , at least when U is obtained from X by removing a single k-rational point [cf. Theorem 3.1]. This, along with the theory of §2, allows one to verify the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama] [cf. Theorem 3.2]. Acknowledgements. I would like to thank Akio Tamagawa, Makoto Matsumoto, and Seidai Yasuda for various useful comments. Also, I would like to thank Yuichiro Hoshi for his careful reading of an earlier version of this manuscript, which led to the discovery of various errors in this earlier version. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 0. 3 Notations and Conventions Numbers:  the profinite completion of the additive group of We shall denote by Z rational integers Z. If p is a prime number, then Z p denotes the ring of p- adic integers; Q p denotes its quotient field. We shall refer to as a p-adic local field (respectively, nonarchimedean local field) any finite field extension of Q p (respectively, a p-adic local field, for some p). A number field is defined to be a finite extension of the field of rational numbers. If Σ is a set of prime numbers, then we shall refer to a positive integer each of whose prime factors belongs to Σ as a Σ-integer. We shall refer to a finite étale covering of schemes whose degree is a Σ-integer as a Σ-covering. Also, we shall write Primes for the set of all prime numbers. Topological Groups: Let G be a Hausdorff topological group, and H ⊆ G a closed subgroup. Let us write G ab for the abelianization of G [i.e., the quotient of G by the closed subgroup of G topologically generated by the commutators of G]. Let us write def Z G (H) = {g ∈ G | g · h = h · g, ∀ h ∈ H} for the centralizer of H in G; N G (H) = {g ∈ G | g · H · g −1 = H} def for the normalizer of H in G; and C G (H) = {g ∈ G | (g · H · g −1 ) def  H has finite index in H, g · H · g −1 } for the commensurator of H in G. Note that: (i) Z G (H), N G (H) and C G (H) are subgroups of G; (ii) we have inclusions H, Z G (H) ⊆ N G (H) ⊆ C G (H) and (iii) H is normal in N G (H). If H = N G (H) (respectively, H = C G (H)), then we shall say that H is normally terminal (respectively, commensurably terminal) in G. Note that Z G (H), N G (H) are always closed in G, while C G (H) is not necessarily closed in G. If G 1 , G 2 are Hausdorff topological groups, then an outer homomorphism G 1 → G 2 is defined to be an equivalence class of continuous homomorphisms G 1 → G 2 , where two such homomorphisms are considered equivalent if they differ by composition with an inner automorphism of G 2 . The group of outer 4 Shinichi Mochizuki automorphisms of G [i.e., bijective bicontinuous outer homomorphisms G → G] will be denoted Out(G). If G is center-free, then there is a natural exact sequence: 1 → G → Aut(G) → Out(G) → 1 [where the homomorphism G → Aut(G) is defined by letting G act on G by conjugation]. If G is a profinite group such that, for every open subgroup H ⊆ G, we have Z G (H) = {1}, then we shall say that G is slim. One verifies immediately that G is slim if and only if every open subgroup of G is center-free [cf. [Mzk5], Remark 0.1.3]. If G is a profinite group and Σ is set of prime numbers, then we shall say that G is a pro-Σ group if the order of every finite quotient group of G is a Σ-integer. If Σ = {l} is of cardinality one, then we shall refer to a pro-Σ group as a pro-l group. Curves: Suppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves of genus g X → S is defined to be a smooth, proper, geometrically connected morphism of schemes X → S whose geometric fibers are curves of genus g. Suppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote the moduli stack of r-pointed stable curves of genus g (where we as- sume the points to be unordered) by M g,r [cf. [DM], [Knud] for an exposition of the theory of such curves; strictly speaking, [Knud] treats the finite étale covering of M g,r determined by ordering the marked points]. The open sub- stack M g,r ⊆ M g,r of smooth curves will be referred to as the moduli stack of smooth r-pointed stable curves of genus g or, alternatively, as the moduli stack of hyperbolic curves of type (g, r). A family of hyperbolic curves of type (g, r) X → S is defined to be a morphism which factors X → Y → S as the composite of an open immersion X → Y onto the complement Y \D of a relative divisor D ⊆ Y which is finite étale over S of relative degree r, and a family Y → S of curves of genus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up to canonical isomorphism. (Indeed, when S is the spectrum of a field, this fact is well-known from the elementary theory of algebraic curves. Next, we consider an arbitrary connected normal S on which a prime l is invertible (which, by Zariski localization, we may assume without loss of generality). Denote by S  → S the finite étale covering parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian of Y . Note that Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 5 S  → S is independent of the choice of (Y, D), since (by the normality of S), S  may be constructed as the normalization of S in the function field of S  (which is independent of the choice of (Y, D) since the restriction of (Y, D) to the generic point of S has already been shown to be unique). Thus, the uniqueness of (Y, D) follows by considering the classifying morphism (associated to (Y, D)) from S  to the finite étale covering of (M g,r ) Z[ 1 l ] parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian [since this covering is well-known to be a scheme, for l sufficiently large].) We shall refer to Y (respectively, D; D) as the compactification (respectively, divisor of cusps; divisor of marked points) of X. A family of hyperbolic curves X → S is defined to be a morphism X → S such that the restriction of this morphism to each connected component of S is a family of hyperbolic curves of type (g, r) for some integers (g, r) as above. A family of hyperbolic curves X → S of type (0, 3) will be referred to as a tripod. If X is a hyperbolic curve over a field K with compactification X ⊆ X, then we shall write X cl ; X cl+ for the sets of closed points of X and X, respectively. If X K (respectively, Y L ) is a hyperbolic curve over a field K (respectively, L), then we shall say that X K is isogenous to Y L if there exists a hyperbolic curve Z M over a field M together with finite étale morphisms Z M → X K , Z M → Y L . Note that in this situation, the morphisms Z M → X K , Z M → Y L induce finite separable inclusions of fields K → M , L → M . [Indeed, this follows immediately from  the easily verified fact that every subgroup G ⊆ × ) such that G {0} determines a field is necessarily contained in M × .] Γ(Z, O Z If X is a generically scheme-like algebraic stack [i.e., an algebraic stack which admits a “scheme-theoretically” dense open that is isomorphic to a scheme] over a field K of characteristic zero that admits a [surjective] finite étale [or, equivalently, finite étale Galois] covering Y → X, where Y is a hyper- bolic curve over a finite extension of K, then we shall refer to X as a hyperbolic orbicurve over K. [Although this definition differs from the definition of a “hy- perbolic orbicurve” given in [Mzk6], Definition 2.2, (ii), it follows immediately from a theorem of Bundgaard-Nielsen-Fox [cf., e.g., [Namba], Theorem 1.2.15, p. 29] that these two definitions are equivalent.] If X → Y is a dominant morphism of hyperbolic orbicurves, then we shall refer to X → Y as a par- tial coarsification morphism if the morphism induced by X → Y on associated coarse spaces [cf., e.g., [FC], Chapter I, §4.10] is an isomorphism. Let X be a hyperbolic orbicurve over an algebraically closed field of char- acteristic zero; denote its étale fundamental group by Δ X . We shall refer to the order of the [manifestly finite!] decomposition group of a closed point x of X as the order of x. We shall refer to the [manifestly finite!] least common multiple of the orders of the closed points of X as the order of X. Thus, it follows immediately from the definitions that X is a hyperbolic curve if and only if the order of X is equal to 1. 6 1. Shinichi Mochizuki Maximal Abelian Cuspidalizations Let X be a proper hyperbolic curve over a field k which is either finite or nonarchimedean local. Write d k for the cohomological dimension of k. Thus, if k is finite (respectively, nonar- chimedean local), then d k = 1 (respectively, d k = 2 [cf., e.g., [NSW], Chapter 7, Theorem 7.1.8, (i)]). If k is finite (respectively, nonarchimedean local), we shall denote the characteristic of k (respectively, of the residue field of k) by p and the number p (respectively, 1) by p † . Also, we shall write Primes † = Primes\(Primes def  {p † }) [where Primes is the set of all prime numbers [cf. §0]; the intersection is taken in the “ambient set” Z]. Let Σ be a set of prime numbers that contains at least one prime number that is invertible in k. Write   def def Σ  = Σ\(Σ {p}); Σ † = Σ\(Σ {p † })   the [where the intersections are taken in the “ambient set” Z]. Denote by Z  † †  and by Z  the maximal pro-Σ quotient of Z.  maximal pro-Σ quotient of Z If k is an algebraic closure of k, then we shall denote the result of base- changing objects over k to k by means of a subscript “k”. Any choice of a basepoint of X determines an algebraic closure k of k, and hence an exact sequence 1 → π 1 (X k ) → π 1 (X) → G k → 1 def where G k = Gal(k/k); π 1 (X), π 1 (X k ) are the étale fundamental groups of X, X k , respectively. Write Δ X for the maximal pro-Σ quotient of π 1 (X k ) and def Π X = π 1 (X)/Ker(π 1 (X k )  Δ X ). Thus, we have an exact sequence: 1 → Δ X → Π X → G k → 1 def Similarly, if we write X × X = X × k X, then we obtain [by considering the maximal pro-Σ quotient of π 1 ((X × X) k )] an exact sequence 1 → Δ X×X → Π X×X → G k → 1 where Π X×X (respectively, Δ X×X ) may be identified with Π X × G k Π X (respec- tively, Δ X × Δ X ). Let Π Z ⊆ Π X×X be an open subgroup that surjects onto def G k . Write Z → X × X for the corresponding covering; Δ Z = Ker(Π Z  G k ). 7 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves Proposition 1.1. (Group-theoreticity of Étale Cohomology) Let †  Z  A be a finite quotient, and N a finite A-module equipped with a con- tinuous Δ X - (respectively, Π X -; Δ Z -; Π Z -) action. Then for i ∈ Z, the natural homomorphism i (X k , N ) H i (Δ X , N ) → H ét i (respectively, H i (Π X , N ) → H ét (X, N ); i H i (Δ Z , N ) → H ét (Z k , N ); i H i (Π Z , N ) → H ét (Z, N )) is an isomorphism. Proof. In light of the Leray spectral sequence for the surjections Π X  G k , Π Z  Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X], it suffices to verify the asserted isomorphism in the case of Δ X . If Y → X k is a connected finite étale Galois Σ-covering, then the associated Leray spectral sequence has “E 2 -term” given by the cohomology of Gal(Y /X) with coefficients in the étale cohomology of Y and abuts to the étale cohomology of X k . By allowing Y to vary, one then verifies immediately that it suffices to verify that every étale cohomology class of Y [with coefficients in N ] vanishes upon pull-back to some [connected] finite étale Σ-covering Y  → Y . Moreover, by passing to an appropriate Y , one reduces immediately to the case where N = A, equipped with the trivial Π X -action. Then the vanishing assertion in question is a tautology for “H 1 ”; for “H 2 ”, it suffices to take Y  → Y so that the degree of Y  → Y annihilates A [cf., e.g., the discussion at the bottom of [FK], p. 136]. Set: def  † ), Z  † ); M X = Hom Z † (H 2 (Δ X , Z def ⊗d −1 M k = Hom Z † (H d k (G k , M X k ⊗d −1 ), M X k )  † -modules of rank one; M X is isomorphic as a G k - Thus, M k , M X are free Z †  † (1)  (1) [where the “(1)” denotes a “Tate twist” — i.e., G k acts on Z module to Z  † (d − 1). via the cyclotomic character]; M k is isomorphic as a G k -module to Z k [Indeed, this follows from Proposition 1.1; Poincaré duality [cf., e.g., [FK],  Chapter II, Theorem 1.13]; the fact, in the finite field case, that G k ∼ = Z  the well- [together with an easy computation of the group cohomology of Z]; known theory of the cohomology of nonarchimedean local fields [cf., e.g., [NSW], Chapter 7, Theorem 7.2.6].] Remark 1. Note that for any open subgroup Π X  ⊆ Π X [which we think of as corresponding to a finite étale covering X  → X], we obtain a natural isomorphism ∼ M X → M X  8 Shinichi Mochizuki  † ) to the induced morphism on group coho- by applying the functor Hom Z † (−, Z  † ) → H 2 (Δ X  , Z  † ) [where Δ X  def mology H 2 (Δ X , Z = Ker(Π X  → G k )] and di- viding by [Δ X : Δ X  ]. [One verifies easily that this does indeed yield an isomor- phism — cf., e.g., the discussion at the bottom of [FK], p. 136.] Moreover, rela-  † , H 2 (Δ X  , M X  ) ∼  † , tive to the tautological isomorphisms H 2 (Δ X , M X ) ∼ = Z = Z ∼ the isomorphism M X → M X  just constructed induces [via the restriction mor-  † → Z  † given by multiplication phism on group cohomology] the morphism Z  by [Δ X : Δ X  ]. Similarly, if k is the base field of X  , then we obtain a natural isomorphism ∼ M k → M k  ∼ by applying the natural isomorphism M X → M X  just constructed and the dual of the natural pull-back morphism on group cohomology and then dividing by [k  : k] [cf., e.g., [NSW], Chapter 7, Corollary 7.1.4]. Proposition 1.2. (Top Cohomology Modules) (i) There are natural isomorphisms:  † ; H d k (G k , M k ) ∼ = Z  † ; H 2 (Δ X , M X ) ∼ = Z ⊗2 ∼  † H 4 (Δ Z , M X ) = Z ;  † H d k +2 (Π X , M X ⊗ M k ) ∼ = Z ⊗2  † H d k +4 (Π Z , M X ⊗ M k ) ∼ = Z ∼  †  † (ii) There is a unique isomorphism M X → Z (1) such that the image of 1 ∈ Z  † ∼ maps via the composite of the isomorphism Z = H 2 (Δ X , M X ) of (i) with ∼ 2 2 †  (1)) induced by the isomorphism the isomorphism H (Δ X , M X ) → H (Δ X , Z ∼  † M X → Z (1) in question to the [first] Chern class of a line bundle of degree 1 on X k . Proof. Assertion (i) follows from the definitions; the Leray spectral se- quence for the surjections Π X  G k , Π Z  Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X]. Assertion (ii) is immediate from the defini- tions.  †  A be a finite quo- Proposition 1.3. (Duality) For i ∈ Z, let Z tient, and N a finite A-module. (i) Suppose that N is equipped with a continuous G k -action. Then the pairing Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 9 H i (G k , N ) × H d k −i (G k , Hom A (N, M k ⊗ A)) → A determined by the cup product in group cohomology and the natural isomor- phisms of Proposition 1.2, (i), determines an isomorphism as follows: ∼ H i (G k , N ) → Hom A (H d k −i (G k , Hom A (N, M k ⊗ A)), A) (ii) Suppose that N is equipped with a continuous Π X - (respectively, Δ X -; Π Z -; Δ Z -) action. Then the pairing H i (Π X , N ) × H d k +2−i (Π X , Hom A (N, M X ⊗ M k ⊗ A)) → A (respectively, H i (Δ X , N ) × H 2−i (Δ X , Hom A (N, M X ⊗ A)) → A; ⊗2 H i (Π Z , N ) × H d k +4−i (Π Z , Hom A (N, M X ⊗ M k ⊗ A)) → A; ⊗2 ⊗ A)) → A) H i (Δ Z , N ) × H 4−i (Δ Z , Hom A (N, M X determined by the cup product in group cohomology and the natural isomor- phisms of Proposition 1.2, (i), determines an isomorphism as follows: ∼ H i (Π X , N ) → Hom A (H d k +2−i (Π X , Hom A (N, M X ⊗ M k ⊗ A)), A) ∼ (respectively, H i (Δ X , N ) → Hom A (H 2−i (Δ X , Hom A (N, M X ⊗ A)), A); ∼ ⊗2 H i (Π Z , N ) → Hom A (H d k +4−i (Π Z , Hom A (N, M X ⊗ M k ⊗ A)), A); ∼ ⊗2 ⊗ A)), A)) H i (Δ Z , N ) → Hom A (H 4−i (Δ Z , Hom A (N, M X  [to- Proof. Assertion (i) follows immediately from the fact that G k ∼ = Z  gether with an easy computation of the group cohomology of Z] in the finite field case; [NSW], Chapter 7, Theorem 7.2.6, in the nonarchimedean local field case. Assertion (ii) then follows from assertion (i); the Leray spectral sequences associated to Π X  G k , Π Z  Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X]; Proposition 1.1; Poincaré duality [cf., e.g., [FK], Chapter II, Theorem 1.13]. Proposition 1.4. (Automorphisms of Cyclotomic Extensions) (i) We have: H 0 (G k , H 1 (Δ X , M X )) = 0. (ii) There are natural isomorphisms 10 Shinichi Mochizuki ∼ ∼ ∧ ∼ ∼ ∧ H 1 (Π X , M X ) → H 1 (G k , M X ) → (k × ) H 1 (Π Z , M X ) → H 1 (G k , M X ) → (k × ) — where the first isomorphisms in each line are induced by the surjections Π X  G k , Π Z  G k ; the second isomorphisms in each line are induced by the ∧ isomorphism of Proposition 1.2, (ii), and the Kummer exact sequence; (k × ) is the maximal pro-Σ † -quotient of k × . Proof. Assertion (i) follows immediately from the “Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], p. 206] in the finite field case; [Mzk8], Lemma 4.6, in the nonarchimedean local field case. The first isomorphisms of assertion (ii) follow immediately from assertion (i) and the Leray spectral sequences associated to Π X  G k , Π Z  G k ; the sec- ond isomorphisms follow immediately from consideration of the Kummer exact sequence for Spec(k). Definition 1.1. (i) Let H be a profinite group equipped with a homomorphism H → Π X . Then we shall refer to the kernel I H of H → Π X as the cuspidal subgroup of H [rel- ative to H → Π X ]. We shall say that H is cuspidally abelian (respectively, cuspidally pro-Σ ∗ [where Σ ∗ is a set of prime numbers]) [relative to H → Π X ] if I H is abelian (respectively, a pro-Σ ∗ group). If H is cuspidally abelian, then observe that H/I H acts naturally [by conjugation] on I H ; we shall say that H is cuspidally central [relative to H → Π X ] if this action of H/I H on I H is trivial. Also, we shall use similar terminology to the terminology just introduced for H → Π X when Π X is replaced by Δ X , Π X×X , Δ X×X . (ii) Let H be a profinite group; H 1 ⊆ H a closed subgroup. Then we shall refer to as an H 1 -inner automorphism of H an inner automorphism induced by conjugation by an element of H 1 . If H  is also a profinite group, then we shall refer to as an H 1 -outer homomorphism H  → H an equivalence class of homomorphisms H  → H, where two such homomorphisms are considered equivalent if they differ by composition by an H 1 -inner automorphism. If H is equipped with a homomorphism H → G k [cf., e.g., the various groups in- def troduced above], and H 1 = Ker(H → G k ), then we shall refer to an H 1 -inner automorphism (respectively, H 1 -outer homomorphism) as a geometrically in- ner automorphism (respectively, geometrically outer homomorphism). If H is equipped with a structure of extension of some other profinite group H 0 by a finite product H 1 of copies of M X , or, more generally, a projective limit H 1 of such finite products, then we shall refer to an H 1 -inner automorphism (re- spectively, H 1 -outer homomorphism) as a cyclotomically inner automorphism Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 11 (respectively, cyclotomically outer homomorphism). If H is equipped with a homomorphism to Π X , Δ X , Π X×X , or Δ X×X [cf. the situation of (i)], and H 1 is the kernel of this homomorphism, then we shall refer to an H 1 -inner automorphism (respectively, H 1 -outer homomorphism) as a cuspidally inner automorphism (respectively, cuspidally outer homomorphism). Next, let Π X  ⊆ Π X be an open normal subgroup, corresponding to a finite étale Galois covering X  → X. Set def Π Z  = Π X  ×X  · Π X ⊆ Π X×X [where we regard Π X as a subgroup of Π X×X via the diagonal map]; write Z  → X × X for the covering determined by Π Z  . Thus, it is a tautology that the diagonal morphism ι : X → X × X lifts to a morphism ι  : X → Z  which induces the inclusion Π X → Π Z  on fundamental groups. If Z → X × X is a connected finite étale covering arising from an open subgroup of Π X×X , write: def U X×X = (X × X)\ι(X); def U Z = (U X×X ) × (X×X) Z Denote by Δ U X×X the maximal cuspidally [i.e., relative to the natural map to π 1 ((X × X) k )] pro-Σ † quotient of the maximal pro-Σ quotient of the tame fundamental group of (U X×X ) k [where “tame” is with respect to the divisor ι(X) ⊆ X × X] and by Π U X×X the quotient π 1 (U X×X )/Ker(π 1 ((U X×X ) k )  Δ U X×X ); write Π U Z ⊆ Π U X×X for the open subgroup corresponding to the finite étale covering U Z → U X×X . Proposition 1.5. (Characteristic Class of the Diagonal) (i) The pull-back morphism arising from the natural inclusion Π X → Π Z  (⊆ Π X×X = Π X × G k Π X ) composed with the natural isomorphism of Proposition 1.2, (i), determines a homomorphism ∼  † H d k +2 (Π Z  , M X ⊗ M k ) → H d k +2 (Π X , M X ⊗ M k ) → Z hence [by Proposition 1.3, (ii)] a class diag η Z ∈ H 2 (Π Z  , M X )  12 Shinichi Mochizuki which is equal to the étale cohomology class associated to ι  (X) ⊆ Z  , or, alter- natively, the [first] Chern class of the line bundle O Z  (ι  (X)). (ii) Denote by   L × diag [Z ] → Z the complement of the zero section in the geometric line bundle [i.e., G m -torsor] determined by O Z  (ι  (X)), by Δ L × [Z  ] the maximal cuspidally pro-Σ † quotient diag  of the maximal pro-Σ quotient of the tame fundamental group of (L × diag [Z ]) k [where “tame” is with respect to the divisor determined by the complement of  1 the G m -torsor L × diag [Z ] in the naturally associated P -bundle], and by Π L × [Z  ] ×   the quotient π 1 (L × diag [Z ])/Ker(π 1 ((L diag [Z ]) k )  Δ L × diag  ). Then [in light of diag [Z ] the isomorphism of Proposition 1.2, (ii)] we have a natural exact sequence 1 → M X → Π L ×  diag [Z ] → Π Z  → 1 diag whose associated extension class is equal to the class η Z  . (iii) The global section of O Z  (ι  (X)) over Z  determined by the natural inclu- sion O Z  → O Z  (ι  (X)) defines a morphism  U Z  → L × diag [Z ] over Z  which induces a surjective homomorphism of groups over Π Z  : Π U Z   Π L ×  diag [Z ] Proof. Assertion (i) follows immediately from Propositions 1.1, 1.2, 1.3, together with well-known facts concerning Chern classes and associated cycles in étale cohomology [cf., e.g., [FK], Chapter II, Definition 1.2, Proposition 2.2]. Assertion (ii) follows from Proposition 1.1; [Mzk7], Definition 4.2, Lemmas 4.4, 4.5. Assertion (iii) follows from [Mzk8], Lemma 4.2, by considering fibers over one of the two natural projections Π Z  → Π X×X  Π X . [Here, we note that although in [Mzk7], §4; [Mzk8], the base field is assumed to be of characteristic zero, one verifies immediately that the same arguments as those applied in loc. cit. yield the corresponding results in the finite field case — so long as we restrict the coefficients of the cohomology modules in question to modules over  † .] Z Definition 1.2. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 13 (i) We shall refer to a covering Z  → X × X as in the above discussion as the diagonal covering associated to the covering X  → X. We shall refer to an extension of profinite groups 1 → M X → D  → Π Z  → 1 diag whose associated extension class is the class η Z of Proposition 1.5, (i), as a  fundamental extension [of Π Z  ]. In the following (ii) — (iv), we shall assume that 1 → M X → D → Π X×X → 1 is a fundamental extension. (ii) Let x, y ∈ X(k); write D x , D y ⊆ Π X for the associated decomposition groups [which are well-defined up to conjugation by an element of Δ X — cf. Remark 2 below]. Now set: def def D x = D| D x × Gk Π X ; D x,y = D| D x × Gk D y Thus, D x (respectively, D x,y ) is an  extension of Π X (respectively, G k ) by M X .  Similarly, if D = i m i · x i , E = j n j · y j are divisors on X supported on points that are rational over k, then set:   def def D D = m i · D x i ; D D,E = m i · n j · D x i ,y j i i,j [where the sums are to be understood as sums of extensions of Π X or G k by M X — i.e., the sums are induced by the additive structure of M X ]. Also, we def shall write C = −D| Π X [where we regard Π X as a subgroup of Π X×X via the diagonal map]. [Thus, C is an extension of Π X by M X whose extension class is the Chern class of the canonical bundle of X.] (iii) Let S ⊆ X(k) be a finite subset. Then we shall write def D S =  D x x∈S [where the product is to be understood as the fiber product over Π X ]. Thus, D S is an extension of Π X by a product of copies of M X indexed by elements of S. We shall refer to D S as a maximal abelian S-cuspidalization [of Π X at S]. Observe that if T ⊆ X(k) is a finite subset such that S ⊆ T , then we obtain a natural projection morphism D T → D S . (iv) We shall refer to a homomorphism Π U X×X → D over Π X×X as a fundamental section if, for some isomorphism of D with Π L × diag that induces the identity on Π X×X , M X , the resulting composite homomor- phism Π U X×X → Π L × is the homomorphism of Proposition 1.5, (iii). diag 14 Shinichi Mochizuki Remark 2. Relative to the situation in Definition 1.2, (ii), conjuga- tion by elements δ ∈ Δ X induces isomorphisms between the different possible choices of “D x ”, all of which lie over the isomorphism between any of these choices and G k induced by the projection Π X  G k . Moreover, by lifting (δ, 1) ∈ Δ X×X ⊆ Π X×X to an element δ D ∈ D, and conjugating by δ D , we obtain natural isomorphisms between the various resulting “D x ’s” which in- duce the identity on the quotient group D x  Π X , as well as on the subgroup M X ⊆ D x . Note that this last property [i.e., of inducing the identity on Π X , M X ] holds precisely because we are working with δ ∈ Δ X ⊆ Π X , as opposed to an arbitrary “δ ∈ Π X ”. Remark 3. By Proposition 1.4, (ii), if E is any profinite group exten- sion of Π X (respectively, G k ; an open subgroup Π Z ⊆ Π X×X that surjects onto G k ) by M X , then the group of cyclotomically outer automorphisms of the extension E [i.e., that induce the identity on Π X (respectively, G k ; Π Z ) and ∧ M X ] may be naturally identified with (k × ) . In particular, in the context of Definition 1.2, (iv), any two fundamental sections of D differ, up to composition ∧ with a cyclotomically inner automorphism of D, by a “(k × ) -multiple”. Next, if k is nonarchimedean local, then set G † k = G k ; if k is finite, then  † ]. Also, we write G † k ⊆ G k for the maximal pro-Σ † subgroup of G k [so G † k ∼ = Z shall use the notation def Π † (−) = Π (−) × G k G † k ⊆ Π (−) def [where “(−)” is any smooth, geometrically connected scheme over k, with arith- metic fundamental group Π (−)  G k ]. Proposition 1.6. idalizations) Let (Basic Properties of Maximal Abelian Cusp- 1 → M X → D → Π X×X → 1 be a fundamental extension; φ : Π U X×X  D a fundamental section; S ⊆ X(k) a finite subset. Then: (i) The profinite groups Δ X×X , Δ X , as well as any profinite group extension of Π † X×X or Π † X by a [possibly empty] finite product of copies of M X is slim [cf. §0]. In particular, the profinite group D S † = D S × G k G † k is slim. def def (ii) For x ∈ X(k), write U x = X\{x}. Denote by Δ U x the maximal cuspidally [i.e., relative to the natural map to π 1 ((U x ) k )] pro-Σ † quotient of the maxi- mal pro-Σ quotient of the tame fundamental group of (U x ) k [where “tame” is Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 15 with respect to the complement of U x in X] and by Π U x the quotient given by π 1 (U x )/Ker(π 1 ((U x ) k )  Δ U x ). Then the inverse image via either of the natu- ral projections Π U X×X  Π X of the decomposition group D x ⊆ Π X is naturally isomorphic to Π U x . In particular, Δ U X×X , Π † U X×X are slim. (iii) For S ⊆ X(k) a finite subset, write:  def U S = U x x∈S [where the product is to be understood as the fiber product over X]. Denote by Δ U S the maximal cuspidally [i.e., relative to the natural map to π 1 ((U S ) k )] pro-Σ † quotient of the maximal pro-Σ quotient of the tame fundamental group of (U S ) k [where “tame” is with respect to the complement of U S in X], and by Π U S the quotient π 1 (U S )/Ker(π 1 ((U S ) k )  Δ U S ). Then Δ U S , Π † U S are slim. Forming the product of the specializations of φ to the various D x × G k Π X ⊆ Π X×X yields homomorphisms  Π U x → D S Π U S → x∈S [where the product is to be understood as the fiber product over Π X ]. Moreover, the composite morphism Π U S → D S is surjective; the resulting quotient of def Δ U S = Ker(Π U S  G k ) is the maximal cuspidally central quotient of Δ U S , relative to the surjection Δ U S  Δ X . def (iv) The quotient of Δ U X×X = Ker(Π U X×X  G k ) determined by φ : Π U X×X  D is the maximal cuspidally central quotient of Δ U X×X , relative to the surjection Δ U X×X  Δ X×X . Proof. Assertion (i) follows immediately from the slimness of Π † X , Δ X [cf., e.g., [Mzk5], Theorem 1.1.1, (ii); the proofs of [Mzk5], Lemmas 1.3.1, 1.3.10], together with the [easily verified] fact that G † k acts faithfully on M X via the cyclotomic character. Next, we consider assertion (ii). The portion of assertion (ii) concerning Π U x follows immediately from the existence of the “homotopy exact sequence associated to a family of curves” [cf., e.g., [Stix], Proposition 2.3]. The slimness assertion then follows from assertion (i) [applied to Π † X ] and the slimness of Δ U x [cf. the proofs of [Mzk5], Lemmas 1.3.1, 1.3.10]. As for assertion (iii), the slimness of Δ U S , Π † U S follows via the arguments given in the proofs of  [Mzk5], Lemmas 1.3.1, 1.3.10. The existence of homomor- phisms Π U S → x∈S Π U x → D S as asserted is immediate from the definitions, assertion (ii). For x ∈ S, write D x [U S ] ⊆ Π U S 16 Shinichi Mochizuki for the decomposition group of x; I x [U S ] ⊆ D x [U S ] for the inertia subgroup. Now it is immediate from the definitions that I x [U S ] maps isomorphically onto the copy M X in D S corresponding to the point x. This implies the desired surjectivity. Since, moreover, it is immediate from the definitions that the cuspidal subgroup of any cuspidally central quotient of Δ U S is generated by the image of the I x [U S ], as x ranges over the elements of S, the final assertion concerning the maximal cuspidally central quotient of Δ U S follows immediately. Assertion (iv) follows by a similar argument to the argument applied to the final portion of assertion (iii). Next, let Z  → X × X (respectively, Z  → X × X; Z ∗ → X × X) be the diagonal covering associated to a covering X  → X (respectively, X  → X; X ∗ → X) arising from an open subgroup of Π X ; denote by ι  : X → Z  (respectively, ι  : X → Z  ; ι ∗ : X → Z ∗ ) the tautological lifting of the diagonal embedding ι : X → X × X and by k  (respectively, k  ; k ∗ ) the extension of k determined by X  (respectively, X  ; X ∗ ). Assume, moreover, that the covering X  → X factors as follows: X  → X  → X ∗ → X Thus, we obtain a factorization Z  → Z  → Z ∗ → X × X. Let 1 → M X → D  → Π Z  → 1 be a fundamental extension of Π Z  . Write  1 → M X → D X  ×X  → Π X  ×X  → 1 for the pull-back of the extension D  via the inclusion Π X  ×X  ⊆ Π Z  . Now if we think of Π X×X or Π X  ×X  as only being defined up to Δ X  × {1}-inner automorphisms, then it makes sense, for δ ∈ Δ X /Δ X  to speak of the pull-back  of the extension D X  ×X  via δ × 1:  1 → M X → (δ × 1) ∗ D X  ×X  → Π X  ×X  → 1 In particular, we may form the fiber product over Π X  ×X  :  def  S X  /X ∗ (D  ) X  ×X  = (δ × 1) ∗ D X  ×X  δ∈Δ X ∗ /Δ X  Thus, S X  /X ∗ (D  ) X  ×X  is an extension of Π X  ×X  by a product of copies of M X indexed by Δ X ∗ /Δ X  ; S X  /X ∗ (D  ) X  ×X  admits a tautological Δ X  × {1}-outer [more precisely: a (Δ X  × {1}) × Π X  ×X  S X  /X ∗ (D  ) X  ×X  -outer] action by the finite group Δ X ∗ /Δ X  ∼ = (Δ X ∗ /Δ X  ) × {1}. Moreover, the natural outer action of Gal(X  /X) ∼ = Gal((X  × X  )/Z  ) ∼ = Π X /Π X  on Π X  ×X  [arising from the diagonal embedding Π X → Π Z  ] clearly lifts to an outer action of Gal(X  /X) on S X  /X ∗ (D  ) X  ×X  , which is compatible, 17 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves relative to the natural action of Gal(X  /X) on Δ X ∗ /Δ X  by conjugation, with the Δ X  × {1}-outer action of Δ X ∗ /Δ X  on S X  /X ∗ (D  ) X  ×X  . Thus, in summary, the natural isomorphism (Δ X ∗ /Δ X  ) × {1}  Gal(X  /X) ∼ = Gal((X  × X  )/Z ∗ ) determines a homomorphism Gal((X  × X  )/Z ∗ ) → Out(S X  /X ∗ (D  ) X  ×X  ) via which we may pull-back the extension “1 → (−) → Aut(−) → Out(−) → 1” [cf. §0; Proposition 1.6, (i)] for S X  /X ∗ (D  ) X  ×X  to obtain an extension  M X → S X  /X ∗ (D  ) → Π Z ∗ → 1 1 → Δ X ∗ /Δ X  in which Π Z ∗ is only determined up to Δ X  × {1}-inner automorphisms. Note, moreover, that every cyclotomically outer automorphism of the extension D  ∧ — i.e., an element of (k × ) [cf. Remark 3] — induces a cyclotomically outer automorphism of S X  /X ∗ (D  ). In particular, we have a natural cyclotomically ∧ outer action of (k × ) on S X  /X ∗ (D  ). Next, let us push-forward the extension S X  /X ∗ (D  ) just constructed via the natural surjection   M X  M X Δ X ∗ /Δ X  Δ X ∗ /Δ X  [which induces the identity morphism M X → M X between the various factors of the domain and codomain], so as to obtain an extension Tr X  /X  :X ∗ (D  ) as follows:  1 → M X → Tr X  /X  :X ∗ (D  ) → Π Z ∗ → 1 Δ X ∗ /Δ X  [in which Π Z ∗ is only determined up to Δ X  × {1}-inner automorphisms]. Proposition 1.7. of the discussion above: (Symmetrizations and Traces) In the notation (i) The extension Tr X  /X  :X  (D  ) of Π Z  by M X is a fundamental extension of Π Z  . (ii) There is a natural commutative diagram:  1 −→ M X −→ S X  /X (D  ) −→ Π X×X −→ 1 Δ X /Δ X  ⏐ ⏐ 1 −→  Δ X /Δ X  ⏐ ⏐ M X −→ S X  /X (Tr X  /X  :X  (D  )) ⏐ ⏐ id −→ Π X×X −→ 1 18 Shinichi Mochizuki [which is well-defined up to Δ X  × {1}-inner automorphisms — cf. Remark 4 below]. (iii) Relative to the commutative diagram of (ii), the natural cyclotomically ∧ ∧ outer action of (k × ) on S X  /X (D  ) lies over the composite of the map (k × ) → ∧ (k × ) given by raising to the [Δ X  : Δ X  ]-power with the natural cyclotom- ∧ ically outer action of (k × ) on S X  /X (Tr X  /X  :X  (D  )). In particular, if N is a positive integer that divides [Δ X  : Δ X  ], then the natural cyclotomically ∧ outer action of an element of (k × ) on S X  /X (D  ) lies over the cyclotomically ∧ outer action of an element of {(k × ) } N on S X  /X (Tr X  /X  :X  (D  )). Proof. To verify assertion (i), observe that it is immediate from the defi- nitions that ι  (X) × Z  (X  × X  ) ⊆ X  × X  is equal to the Δ X  /Δ X  -orbit of ι  (X) × Z  (X  × X  ) ⊆ X  × X  . Now assertion (i) follows by translating this observation into the language of étale cohomology classes associated to subvarieties; assertions (ii), (iii) follow for- mally from assertion (i) and the definitions of the various objects involved. Remark 4. Relative to the commutative diagram of Proposition 1.7, (ii), note that, although S X  /X (Tr X  /X  :X  (D  )) is, by definition, only well- defined up to Δ X  × {1}-inner automorphisms, the push-forward of S X  /X (D  ) by   M X → M X Δ X /Δ X  Δ X /Δ X  is well-defined up to Δ X  × {1}-inner automorphisms. That is to say, the push- forward extension implicit in this commutative diagram furnishes a canonically more rigid version of the extension S X  /X (Tr X  /X  :X  (D  )). Definition 1.3. (i) We shall refer to the extension S X  /X ∗ (D  ) [of Π Z ∗ ] constructed from the fundamental extension D  as the [X  /X ∗ -]symmetrization of D  , or, alterna- tively, as a symmetrized fundamental extension. We shall refer to the extension Tr X  /X  :X ∗ (D  ) [of Π Z ∗ ] constructed from the fundamental extension D  as the [X  /X  : X ∗ -]trace of D  , or, alternatively, as a trace-symmetrized funda- mental extension. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 19 (ii) If D  is a fundamental extension of Π Z  , then we shall refer to as a morphism of trace type any morphism S X  /X (D  ) → S X  /X (D  ) obtained by composing the morphism S X  /X (D  ) → S X  /X (Tr X  /X  :X  (D  )) of Proposition 1.7, (ii), with a morphism S X  /X (Tr X  /X  :X  (D  )) → S X  /X (D  ) arising [by the functoriality of the construction of “S X  /X (−)”] from an isomor- ∼ phism of [fundamental] extensions Tr X  /X  :X  (D  ) → D  of Π Z  by M X [which induces the identity on Π Z  , M X ]. (iii) We shall refer to as a pro-symmetrized fundamental extension any compat- ible system [indexed by the natural numbers] . . .  S i  . . .  S j  . . .  Π X×X of morphisms of trace type [up to inner automorphisms of the appropriate type] between symmetrized fundamental extensions, where S i is the X i /X- symmetrization of a fundamental extension of Π Z i ; Z i is the diagonal covering associated to an open normal subgroup Π X i ⊆ Π X ; the intersection of the Π X i is trivial. In this situation, we shall refer to the inverse limit profinite group def S i S ∞ = lim ←− i as the limit of the pro-symmetrized fundamental extension {S i }; any profinite group S ∞ arising in this fashion will be referred to as a pro-fundamental exten- sion [of Π X×X ]. (iv) Let S ⊆ X(k) be a finite subset; S  an X  /X-symmetrization of a funda- mental extension D  of Π Z  . Then we shall write  def  S S  = S D x × G Π X x∈S k [where the product is to be understood as the fiber product over Π X ]. Thus, S S  is an extension of Π X by a product of copies of M X . Similarly, given a projective system {S i } as in (iii), we obtain a projective system {(S i ) S }, with inverse limit: (S ∞ ) S We shall refer to (S ∞ ) S as a maximal abelian S-pro-cuspidalization [of Π X at S]. Observe that if T ⊆ X(k) is a finite subset such that S ⊆ T , then we obtain a natural projection morphism (S ∞ ) T → (S ∞ ) S . 20 Shinichi Mochizuki Remark 5. Let D be as in Definition 1.2, (iii); S  , {S i }, S ∞ as in Definition 1.3, (iii), (iv). Then observe that it follows from Proposition 1.6, (i), † [i.e., the result of applying that the “daggered versions” D † , (S  ) † , S i † , and S ∞ †  “× G k G k ” to D, S , S i , and S ∞ ] are slim. In particular, if S ⊆ X cl is any finite set of closed points of X, then we may form the objects D S † ; (S  ) † S ; (S i ) † S ; (S ∞ ) † S by passing to a Galois covering X k S → X [i.e., the result of base-changing X to some finite Galois extension k S of k] such that the closed points of X k S that lie over points of S are rational over k S ; forming the various objects in question over X k S [cf. Definition 1.2, (iii); Definition 1.3, (iv)]; and, finally, “descending to X” via the natural outer action of G k /G † k S on the various objects in question [cf. the exact sequence “1 → (−) → Aut(−) → Out(−) → 1” of §0; the slimness mentioned above]. Thus, in the remainder of this paper, we shall often speak of the various objects defined in Definition 1.2, (iii); Definition 1.3, (iv), even when the points of the finite set S are not necessarily rational over k. Before proceeding, we note the following: Lemma 1.1. (Conjugacy Estimate) Let H ⊆ Δ X be a normal open subgroup; a ∈ Δ X /H an element not equal to the identity; N a Σ † -integer [cf. §0]. Then there exists a normal open subgroup H  ⊆ Δ X contained in H such that for any normal open subgroup H  ⊆ Δ X contained in H  and any a  ∈ Δ X /H  that lifts a, the cardinality of the H-conjugacy class Conj(a  , H  ) ⊆ Δ X /H  of a  in Δ X /H  is divisible by N . Proof. In the notation of the statement of Lemma 1.1, let us denote by Z(a  , H  ) ⊆ H the subgroup of elements δ ∈ H such that δ · a  · δ −1 = a  in Δ X /H  . Then it is immediate that if a  is the image of a  in Δ X /H  , then Z(a  , H  ) ⊆ Z(a  , H  ), so the cardinality of Conj(a  , H  ) ∼ = H/Z(a  , H  ) is   ∼   divisible by the cardinality of Conj(a , H ) = H/Z(a , H ). Thus, it suffices to find a normal open subgroup H  ⊆ H such that for any a  ∈ Δ X /H  that lifts a, the cardinality of Conj(a  , H  ) is divisible by N . To this end, let us consider, for some prime number l ∈ Σ † , the maximal pro-l quotient H[l] of the abelianization H ab of H. Note that Δ X /H acts by conjugation on H ab , H[l]. Now I claim that there exists a [nonzero] h l ∈ H[l] such that a(h l ) = h l . Indeed, if this claim were false, then it would follow that a acts trivially on H[l]. But since a induces a nontrivial automorphism of the covering of X k determined by H, it follows that a induces a nontrivial automorphism of the l-power torsion points of the Jacobian of X k [since these points are Zariski dense in this Jacobian] — a contradiction. This completes the proof of the claim. 21 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves Now let j ∈ H be an element that lifts the various h l obtained above for the [finite collection of] primes l that divide N ; let a X ∈ Δ X be an element that lifts a. Then observe that for some integer power M of N that is independent ab ⊗ (Z/M Z) is of the choice of a X , the image of j n · a X · j −n · a −1 X in H    nonzero, for n ∈ Z with nonzero image in Z/N · Z. Thus, if we take H  equal to the inverse image of M · H ab in H(⊆ Δ X ), we obtain that the intersection of the subgroup j Z ⊆ H with Z(a  , H  ) [where a  ∈ Δ X /H  lifts a] does not  with nonzero image in Z/N   But this implies that the contain j n , for n ∈ Z · Z. Z   N · Z , hence that [H : Z(a  , H  )] is divisible by intersection (j ) Z(a , H ) ⊆ j N , as desired. Next, we consider the following fundamental extensions of Π Z  , Π Z  : D  = Π L × def diag [Z  ] ; D  = Tr X  /X  :X  (D  ) def [cf. Proposition 1.5, (ii)]. Note that in this situation, it follows immediately ∼ from the definitions that we obtain a natural isomorphism D  → Π L × [Z  ] , diag which we shall use in the following discussion to identify D  , Π L × [Z  ] . Thus, diag we have fundamental sections: Π U Z   D  ; Π U Z   D  [cf. Proposition 1.5, (iii)]. In particular, by pulling back from Z  to X  × X  , we obtain a surjection: Π U X  ×X   D  X  ×X  Now if we apply the natural outer (Δ X /Δ X  ) × {1}-action on Π U X  ×X  to this surjection, it follows from the definition of “S X  /X (D  )” that we obtain a natural homomorphism Π U X  ×X   S X  /X (D  ) X  ×X  which is easily verified [cf. Proposition 1.6, (ii), (iii)] to be surjective. Since, moreover, the construction of this surjective homomorphism is manifestly com- patible with the outer actions of Gal(X  /X) on both sides, we thus obtain a natural surjection: Π U X×X  S X  /X (D  ) Now let us denote by D X ⊆ Π U X×X the decomposition group of the subvariety ι(X) ⊆ X × X. [Thus, D X is well- defined up to conjugation; here, we assume that we have chosen a conjugate that maps to the image of the diagonal embedding Π X → Π X×X via the natural surjection Π U X×X  Π X×X .] Observe that we have a natural exact sequence 1 → I X → D X → Π X → 1 22 Shinichi Mochizuki [where I X — i.e., the inertia subgroup of D X — is defined so as to make the sequence exact], together with a natural isomorphism I X ∼ = M X . Also, we shall def def write D X  = D X Π U X  ×X  ; D X  = D X Π U X  ×X  . Since the construction just carried out for double primed objects may also be carried out for single primed objects, we thus obtain the following: Proposition 1.8. (Symmetrized Fundamental Sections) In the notation of the discussion above: (i) There is a natural commutative diagram: D X ⊆  S X  /X (D  ) ⏐ ⏐ ⏐ ⏐ id D X Π U X×X ⏐ ⏐ id ⊆ Π U X×X  S X  /X (D  ) [where the vertical arrow on the right is the morphism in the diagram of Propo- sition 1.7, (ii)]. (ii) Denote by means of a subscript X  the result of pulling back extensions of Π X×X , Π Z  , Π X  ×X  to Π X  [via the diagonal inclusion]. Then the projection [cf. the fiber product defining S X  /X (D  )] to the factor labeled “Δ X  /Δ X  ” detemines a natural surjection ζ  : S X  /X (D  ) X   D  X  whose restriction to D X  [i.e., relative to the arrows in the first line of the ∼ commutative diagram of (i)] defines an isomorphism D X  → D  X  . Moreover, the cuspidal subgroup of D X  maps isomorphically onto the factor of M X in S X  /X (D  ) labeled “Δ X  /Δ X  ”. In particular, if we denote by S X  /X (D  ) = the quotient of S X  /X (D  ) by this factor of M X , then ζ  determines a surjec- tion  : S X  /X (D  ) = ζ = X   Π X  ∼ whose restriction to the quotient D X   Π X  is equal to the identity Π X  → Π X  [up to geometric inner automorphisms]. Thus, we have a natural commuta- tive diagram [well-defined up to geometric inner automorphisms] D X  → ⏐ ⏐ Π X  S X  /X (D  ) X  ζ  −→ D  X  ⏐ ⏐ → S X  /X (D  ) = X  ⏐ ⏐  ζ  = −→ Π X  23 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves in which the two horizontal composites are isomorphisms; the vertical arrows are surjections; both squares are cartesian. (iii) If we carry out the construction of (ii) for the single primed objects, then the commutative diagram of (i) induces a natural commutative diagram [well-defined up to geometric inner automorphisms]: Π X   ζ  = → S X  /X (D  ) = X  −→ Π X  ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Π X  →  ζ  = S X  /X (D  ) = X  −→ Π X  Moreover, there is a natural outer action of Gal(X  /X) (respectively, Gal(X  /X)) on the first (respectively, second) line of this diagram; these outer actions are compatible with one another. (iv) When considered up to cyclotomically inner automorphisms, the sections  of ζ = form a torsor over the group  ((k  ) × ) ∧ (Δ X /Δ X  )\(Δ X  /Δ X  ) [where the “\” denotes the set-theoretic complement]. The Gal(X  /X)-equi-  form a torsor over the Gal(X  /X)-invariant subgroup variant sections of ζ = of this group. Similar statements hold for the single primed objects. (v) The double and single primed torsors of equivariant sections of (iv) are related, via the right-hand square of the diagram of (iii), by a homomorphism  ∧ ((k  ) × ) Gal(X  /X) (Δ X /Δ X  )\(Δ X  /Δ X  ) →  ∧ ((k  ) × ) Gal(X  /X) (Δ X /Δ X  )\(Δ X  /Δ X  ) [where the superscripts denote the result of taking invariants with respect to the action of the superscripted group] that satisfies the following property: An element ξ  of the domain maps to an element of the codomain whose component in the factor labeled a  ∈ Δ X /Δ X  is a product of  ∧ elements of ((k  ) × ) of the form N k   /k  (λ  ) n . a  ∧ Here, a ∈ (Δ X /Δ X  )\(Δ X  /Δ X  ) maps to a  in Δ X /Δ X  ; λ  ∈ ((k  ) × ) is the component of ξ  in the factor labeled a  ; k a   is an intermediate field 24 Shinichi Mochizuki ∧ ∧ extension between k  and k  such that λ  ∈ ((k a   ) × ) ; N k a   /k  : ((k a   ) × ) → ∧ ((k  ) × ) is the norm map; n  is the cardinality of the Δ X  -conjugacy class of a  in (Δ X /Δ X  ). In particular, by Lemma 1.1 [where we take “H” to be Δ X  , “H  ” to be Δ X  ], for a given Δ X  , if, for a given positive integer N , Δ X  is “sufficiently small”, then an arbitrary Gal(X  /X)-equivariant section of   lies over the canonical section of ζ = given in (iii), up to the cyclotomically ζ = outer action of some N -th power of an element of the single primed version of the group exhibited in the display of (iv). Proof. All of these assertions follow immediately from the definitions [and, in the case of assertion (iv), Proposition 1.4, (ii)]. Definition 1.4. Let D  be a fundamental extension of Π Z  ; {S i } a pro- symmetrized fundamental extension, with limit S ∞ [cf. Definition 1.3, (iii)]. (i) We shall refer to as a symmetrized fundamental section a homomorphism Π U X×X  S X  /X (D  ) obtained by composing the surjection Π U X×X  S X  /X (D  ) of Proposition 1.8, ∼ (i), with the isomorphism S X  /X (D  ) → S X  /X (D  ) induced by an isomorphism ∼ D  → D  of fundamental extensions of Π Z  by M X [which induces the identity on Π Z  , M X ]. We shall refer to an inclusion D X → S X  /X (D  ) obtained by restricting a symmetrized fundamental section to D X ⊆ Π U X×X [cf. Proposition 1.8, (i)] as a fundamental inclusion. (ii) We shall refer to a compatible system of symmetrized fundamental sections Π U X×X → S i as a pro-symmetrized fundamental section and to the resulting limit homomorphism Π U X×X → S ∞ as a pro-fundamental section. Similarly, we have a notion of “pro-fundamental inclusions”. Remark 6. Thus, by the above discussion, if we take the “S i ” to be the symmetrizations of the Π L × [Z  ] as in Proposition 1.5, (ii), then we obtain nat- diag ural pro-fundamental sections and pro-fundamental inclusions [cf. Proposition 1.8, (i), (ii), (iii)]. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 25 Proposition 1.9. (Maximal Cuspidally Abelian Quotients) Let {S i } be a pro-symmetrized fundamental extension, with limit S ∞ [cf. Definition 1.3, (iii)] and pro-fundamental section Π U X×X  S ∞ [cf. Defi- nition 1.4, (ii)]; S ⊆ X cl a finite set of closed points [cf. Remark 5]. Then: (i) The pro-fundamental section Π U X×X  S ∞ determines a surjection Π U S  (S ∞ ) S [cf. Proposition 1.6, (iii)]. The resulting quotient of Δ U S (respectively, Π U S ) is the maximal cuspidally abelian quotient Δ U S  Δ c-ab U S (respectively, c-ab Π U S  Π U S ) of Δ U S (respectively, Π U S ). (ii) The quotient of Δ U X×X (respectively, Π U X×X ) induced by the pro-funda- mental section Π U X×X  S ∞ is the maximal cuspidally abelian quotient c-ab [which we shall denote by] Δ U X×X  Δ c-ab U X×X (respectively, Π U X×X  Π U X×X ) of Δ U X×X (respectively, Π U X×X ). Proof. Indeed, this follows as in the proof of Proposition 1.6, (iii), (iv), by observing that the cuspidal subgroup of the maximal cuspidally abelian quotient of Δ U S (respectively, Δ U X×X ) is naturally isomorphic to the inverse limit of the cuspidal subgroups of the maximal cuspidally central quotients of the Δ U S × Δ X Δ X  (⊆ Δ U S ) (respectively, Δ U X  ×X  ) [as Δ X  ⊆ Δ X ranges over the open normal subgroups of Δ X ]. Proposition 1.10. (Automorphisms and Commensurators) Let {S i } be a pro-symmetrized fundamental extension, with limit S ∞ [cf. Definition 1.3, (iii)] and pro-fundamental inclusion D X → S ∞ [cf. Defi- nition 1.4, (ii)]. Then: (i) Any automorphism α of the profinite group Π c-ab U X×X which (a) is compatible with the natural surjection Π c-ab U X×X  Π X×X and induces the identity on Π X×X ; (b) preserves the image of M X ∼ = I X ⊆ D X via the natural inclusion D X → Π c-ab U X×X is cuspidally inner. (ii) Π X (respectively, Δ X ) is commensurably terminal [cf. §0] in Π X×X 26 Shinichi Mochizuki (respectively, Δ X×X ). (iii) D X is commensurably terminal in S i , S ∞ ∼ = Π c-ab U X×X . Proof. First, we verify assertion (i). By Proposition 1.9, (ii), we have a ∼ natural isomorphism Π c-ab U X×X → S ∞ , so we may think of α as an automorphism of S ∞ . In light of (a); Proposition 1.6, (iii), it follows that α is compatible with the natural surjections S ∞  S i . Write α i for the automorphism of S i induced by α. By (a), (b), it follows that α i is an automorphism of the extension S i of Π X×X by a product of copies of M X which induces the identity on both Π X×X and the product of copies of M X [cf. the definition by a certain fiber product of the symmetrized fundamental extension S i ]. [Here, we note that the fact that α i induces the identity on each copy of M X follows by considering the non-torsion [cf. Propositions 1.2, (ii); 1.5, (i), (ii)] extension class determined by that copy of M X [which is preserved by α i !], together with the fact that α i induces the identity on the second cohomology groups of open subgroups of Δ X×X with coefficients in M X .] Thus, up to cyclotomically inner automorphisms, α i arises ∧ from a collection of elements of (k i × ) , where k i is some finite Galois extension of k [cf. Proposition 1.4, (ii)], one corresponding to each copy of M X . Moreover, since these copies of M X are permuted by the action of Π X×X by conjugation, it follows that [up to cyclotomically inner automorphisms] α i arises from a single ∧ ∧ ∧ element of (k i × ) , which in fact belongs to (k × ) (⊆ (k i × ) ) [as one sees by considering the conjugation action via the “G k portion” of Π X×X ]. On the other hand, since the α i form a compatible system of automorphisms of the S i , ∧ it follows from Proposition 1.7, (iii), that this element of (k × ) must be equal to 1, as desired. Next, to verify assertion (ii), let us observe that it suffices to show that Δ X is commensurably terminal in Δ X×X . But this follows immediately from the fact that Δ X is slim [cf. Proposition 1.6, (i)]. Finally, we consider assertion (iii). Clearly, it suffices to show that D X is commensurably terminal in S i . By assertion (ii), to verify this commensurable terminality, it suffices to show that the [manifestly abelian] cuspidal subgroup H i ⊆ S i [i.e., relative to the natural surjection S i  Π X×X ] satisfies the following property: Every h ∈ H i such that h δ − h ∈ D X , for all δ in some open subgroup J of D X , satisfies h ∈ D X . But this property follows immediately [cf. the definition by a certain fiber product of the symmetrized fundamental extension S i ] from the fact that, for J sufficiently small, the J-module H i /(D X H i ) is isomorphic to a direct product of a finite number of copies of M X . The following result is the main result of the present §1: Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 27 Theorem 1.1. (Reconstruction of Maximal Cuspidally Abelian Quotients) Let X, Y be hyperbolic curves over a finite or nonarchimedean local field; denote the base fields of X, Y by k X , k Y , respectively. Let Σ X (respectively, Σ Y ) be a set of prime numbers that contains at least one prime number that is invertible in k X (respectively, k Y ); write Δ X (respectively, Δ Y ) for the maximal cuspidally pro-Σ † X (respectively, pro-Σ † Y ) quotient of the maximal pro-Σ X (respectively, pro-Σ Y ) quotient of the tame fundamen- tal group of X k X (respectively, Y k Y ) [where “tame” is with respect to the complement of X k X (respectively, Y k Y ) in its canonical compactification], and Π X (respectively, Π Y ) for the corresponding quotient of the étale fundamental group of X (respectively, Y ). Let ∼ α : Π X → Π Y be an isomorphism of profinite groups. Then: (i) We have Σ † X = Σ † Y ; write Σ † = Σ † X = Σ † Y . Moreover, k X is a finite field if and only if k Y is; α preserves the decomposition groups of cusps; X is of type (g, r) [where g, r ≥ 0 are integers such that 2g − 2 + r > 0] if and only if Y is of type (g, r). Finally, if k X , k Y are nonarchimedean local, then their residue characteristics coincide. def (ii) α is compatible with the natural quotients Π X  G k X , Π Y  G k Y . (iii) Assume that X, Y are proper. Denote by Π U X×X  Π c-ab U X×X , Π U Y ×Y  Π c-ab U Y ×Y the maximal cuspidally [i.e., relative to the natural surjections Π U X×X  Π X×X , Π U Y ×Y  Π Y ×Y ] abelian quotients [cf. Proposition 1.9]. Then there is a commutative diagram [well-defined up to cuspidally inner auto- morphisms] Π c-ab U X×X α c-ab −→ Π c-ab U Y ×Y ⏐ ⏐ ⏐ ⏐ Π X×X α×α −→ Π Y ×Y — where, the horizontal arrows are isomorphisms which are compatible with c-ab the natural inclusions D X → Π c-ab U X×X , D Y → Π U Y ×Y [cf. Proposition 1.8, (i)]; the vertical arrows are the natural surjections. Finally, the correspondence α → α c-ab is functorial [up to cuspidally inner automorphisms] with respect to α. Proof. First, we consider assertions (i), (ii). Note that k X is finite if and only if, for every open subgroup H ⊆ Π X , the quotient of the abelianization 28 Shinichi Mochizuki H ab by the closure of the torsion subgroup of H ab is topologically cyclic [cf. [Tama], Proposition 3.3, (ii)]; a similar statement holds for k Y , Π Y . Thus, k X is finite if and only if k Y is. Now suppose that k X , k Y are finite. Then assertion (ii) also follows from [Tama], Proposition 3.3, (ii). The fact that Σ † X = Σ † Y then follows from the following observation: The subset Σ † X ⊆ Primes is the subset on which the function Primes l → dim Q l ((Δ X ) ab ⊗ Q l ) attains its maximum value [cf. [Tama], Proposition 3.1]; a similar statement holds for Y . Now by considering the respective outer actions of G k X , G k Y on the maximal pro-l quotients of Δ X , Δ Y , for some l ∈ Σ † , we obtain that α preserves the decomposition groups of cusps [hence that X is of type (g, r) if and only if Y is of type (g, r)], by [Mzk9], Corollary 2.7, (i). This completes the proof of assertions (i), (ii) in the finite field case. Next, let us assume that k X , k Y are nonarchimedean local. Then the portion of assertion (i) concerning Σ X = Σ † X , Σ Y = Σ † Y follows by considering the cohomological dimension of Π X , Π Y — cf., e.g., Proposition 1.3, (ii) [in the def proper case]. As for assertion (ii), if the cardinality of Σ = Σ † is ≥ 2, then assertion (ii) follows from the evident pro-Σ analogue of [Mzk5], Lemma 1.3.8; if the cardinality of Σ is 1, then assertion (ii) follows from Lemma 1.2, (c), (d) below. Now the portion of assertion (i) concerning the residue characteristics of k X , k Y follows from assertion (ii) and [Mzk5], Proposition 1.2.1, (i); the fact that α preserves the decomposition groups of cusps [hence that X is of type (g, r) if and only if Y is of type (g, r)] follows from [Mzk9], Corollary 2.7, (i). This completes the proof of assertions (i), (ii) in the nonarchimedean local field case. Finally, we consider assertion (iii). It follows from the definitions that α ∼  → X, Z Y  → Y are induces an isomorphism M X → M Y . If, moreover, Z X diagonal coverings corresponding to [connected] finite étale Galois coverings X  → X, Y  → Y that arise from open subgroups of Π X , Π Y that correspond via α, then α induces an isomorphism of group cohomology modules ∼ 2  , M X ) → H (Π Z  , M Y ) H 2 (Π Z X Y that preserves the extension classes associated to fundamental extensions of  , Π Z  Π Z X [cf. Proposition 1.5, (i)]. In particular, if D  (respectively, E  ) Y  (respectively, Π Z Y  ), then α induces an is a fundamental extension of Π Z X isomorphism ∼ D  → E  ∼ ∼  which is compatible with the morphisms M X → M Y , Π Z X → Π Z Y  already induced by α, and, moreover, uniquely determined, up to cyclotomically inner ∧ × ∧ ) (respectively, (k Y × ) ) [cf. Proposition automorphisms, and the action of (k X   1.4, (ii)]. On the other hand, by allowing X , Y to vary, taking symmetriza- tions of the fundamental extensions involved [which may be constructed entirely group-theoretically!], and making use of the vertical morphism in the center of Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 29 the diagram of Proposition 1.7, (ii) [again an object which may be constructed entirely group-theoretically!], it follows from Proposition 1.7, (iii), that the inde- ∧ ∼ × ∧ ) , (k Y × ) terminacy of the isomorphism D  → E  arising from the action of (k X ∼ “converges to the identity indeterminacy” [i.e., by taking D  → E  to arise as ∼ just described from an isomorphism of fundamental extensions D  → E  as- sociated to [connected] finite étale coverings X  → X  , Y  → Y  [that arise from open subgroups of Π X , Π Y that correspond via α], where the open sub- groups Π X  ⊆ Π X  , Π Y  ⊆ Π Y  are sufficiently small]. Thus, in light of the manifest functoriality of the vertical morphism in the center of the diagram of Proposition 1.7, (ii) [the detailed explication of which, in terms of various com- mutative diagrams, is a routine task which we leave to the reader!], we obtain an isomorphism ∼ {S i } → {T j } of pro-symmetrized fundamental extensions [cf. Definition 1.3, (iii)] of Π X×X , Π Y ×Y , respectively, which arises from α and is completely determined up to cyclotomically inner automorphisms. Here, we pause to note that although in the construction of the symmetrization of a fundamental extension D  (re- spectively, E  ), one must, a priori, contend with a certain indeterminacy with respect to Δ X  × {1}- (respectively, Δ Y  × {1}-)inner automorphisms [cf., e.g., Proposition 1.7, (ii)], in fact, by allowing X  , Y  to vary, this indeterminacy also “converges to the identity indeterminacy” [cf. Remark 4]. Thus, in summary, α induces an isomorphism [well-defined up to cyclo- tomically [or, alternatively, cuspidally] inner automorphisms] ∼ S ∞ → T ∞ of pro-fundamental extensions of Π X×X , Π Y ×Y , respectively. Moreover, by applying the fact that the left-hand square of the commutative diagram of Proposition 1.8, (ii), is cartesian, together with the fact that the “canonical  ” that appears in Proposition 1.8, (iii), is completely determined section” of “ζ = [cf. Proposition 1.8, (v); Lemma 1.1] by the condition that it lie under an  ” associated arbitrary “equivariant section” [cf. Proposition 1.8, (iv)] of the “ζ = to coverings “X  → X  ” arising from arbitrarily small open subgroups Π X  ⊆ ∼ Π X , it follows that the isomorphism S ∞ → T ∞ just obtained is compatible with the pro-fundamental inclusions D X → S ∞ , D Y → T ∞ . In particular, by Proposition 1.9, (ii) [cf. also Proposition 1.8, (i)], we conclude that α induces an isomorphism [well-defined up to cuspidally inner automorphisms] (S ∞ ∼ =) ∼ c-ab Π c-ab U X×X → Π U Y ×Y ( ∼ = T ∞ ) c-ab which is compatible with the natural inclusions D X → Π c-ab U X×X , D Y → Π U Y ×Y . Finally, the functoriality of this isomorphism follows from the naturality of its construction. 30 Shinichi Mochizuki Remark 7. It follows immediately from the naturality of the construc- tions used in the proof of Theorem 1.1, (iii), that when “α” arises from an ∼ isomorphism of schemes X → Y , the resulting α c-ab of Theorem 1.1, (iii), co- incides with the morphism induced on fundamental groups by the resulting ∼ isomorphism of schemes U X×X → U Y ×Y . Lemma 1.2. (Normal Subgroups of the Absolute Galois Group of a Nonarchimedean Local Field) Let k be a nonarchimedean local field of residue characteristic p; write G k for the absolute Galois group of k. Also, let us write I ⊆ G k for the inertia subgroup of G k and W ⊆ I for the wild inertia subgroup. [Here, we recall that W is the unique Sylow pro-p subgroup of I.] Let H ⊆ G k be a closed subgroup that satisfies [at least] one of the following four conditions: (a) H is a finite group. (b) H commutes with W . (c) H is a pro-prime-to-p group [i.e., the order of every finite quotient group of H is prime to p] that is normal in G k . (d) H is a topologically finitely generated pro-p group that is nor- mal in G k . Then H = {1}. Proof. Indeed, suppose that H satisfies condition (a). Then the fact that H = {1} follows from [NSW], Corollary 12.1.3, Theorem 12.1.7. Now suppose that H satisfies condition (b). Then by the well-known functorial isomorphism [arising from local class field theory] between the additive group underlying a finite field extension of k that corresponds to an open subgroup J ⊆ G k and the tensor product with Q p of the image of W J in the abelianization J ab , it follows immediately that the conjugation action of H on W is nontrivial, whenever H is nontrivial. Thus we conclude again that H = {1}. Next, suppose that H satisfies condition (c). Then since H, W are both normal in G k , it follows [by considering commutators of elements of H with elements of W ] that arbitrary elements of H commute with arbitrary elements of W . In particular, H satisfies condition (b), so we conclude yet again that H = {1}. Finally, we assume that H is nontrivial and satisfies condition (d). Then I claim that H has trivial image Im(H) in G k /W . Indeed, since I/W , Im(H) are normal in G k /W , and, moreover, I/W is pro-prime-to-p, it follows that these two groups commute. On the other hand, since, as is well-known, G k /I acts faithfully [by conjugation, via the cyclotomic character] on I/W , it thus follows that Im(H) is trivial, as asserted. Thus, H ⊆ W . Since [as in well-known — Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 31 cf., e.g., the proof of [Mzk4], Lemma 15.6] W is a free pro-p group of infinite rank, we thus conclude that there exists an open subgroup U ⊆ W [so U is also a free pro-p group of infinite rank] containing H such that the natural map H ab ⊗ F p → U ab ⊗ F p is injective, but not surjective. Then it follows immediately from the well-known theory of free pro-p groups that there exists a set of free topological generators {ξ i } i∈I [so the index set I is infinite] of U such that for some finite subset J ⊆ I, the elements {ξ j } j∈J lie in and topologically generate H. On the other hand, since H is normal in U , it follows from the well-known structure of free pro-p groups that we obtain a contradiction. This completes the proof of Lemma 1.2. Remark 8. The author would like to thank A. Tamagawa for inform- ing him of the content of Lemma 1.2. Definition 1.5. In the situation of Theorem 1.1, (i), (ii), suppose fur- def ther that Σ X = Σ Y ; write Σ = Σ X = Σ Y . (i) If, for every finite étale covering X  → X of X arising from an open subgroup Π X  ⊆ Π X , it holds that the map from (X  ) cl+ [cf. §0] to conjugacy classes of closed subgroups of Π X  given by assigning to a closed point its associated decomposition group is injective, then we shall say that X is Σ-separated. (ii) If the map induced by α on closed subgroups of Π X , Π Y induces a bijection between the decomposition groups of the points of X cl+ , Y cl+ , then we shall say that α is quasi-point-theoretic. If α is quasi-point-theoretic, and, moreover, X, Y are Σ-separated — in which case α induces bijections ∼ X cl → Y cl ; ∼ X cl+ → Y cl+ — then we shall say that α is point-theoretic. (iii) Suppose further that we are in the finite field case. Then we shall say ∼ that α is Frobenius-preserving if the isomorphism G k X → G k Y induced by α [cf. Theorem 1.1, (ii)] maps the Frobenius element of G k X to the Frobenius element of G k Y . 32 Shinichi Mochizuki Remark 9. In the finite field case, when Σ † = Primes † , the Frobenius element of G k X may be characterized as in [Tama], Proposition 3.4, (i), (ii); a similar statement holds for the Frobenius element of G k Y . [Moreover, in the proper case, the Frobenius element of G k X may be characterized as the element of G k X that acts on M X via multiplication by the cardinality of k X , i.e., the cardinality of H 1 (G k X , M X ) plus 1.] Thus, when Σ † = Primes † , any α as in Theorem 1.1, (i), (ii), is automatically Frobenius-preserving. Remark 10. Let us suppose that we are in the situation of Definition 1.5, and that the base fields k X , k Y are finite. Let us refer to as a quasi-section [of Π X  G k X ] any closed subgroup D ⊆ Π X [i.e., such as a decomposition group of a point ∈ X cl ] that maps isomorphically onto an open subgroup of G k X . Let us refer to a quasi-section of Π X  G k X as a subdecomposition group if it is contained in some decomposition group of a point ∈ X cl . Then: (i) Since X is not necessarily Σ-separated, it is not necessarily the case that decomposition groups of points ∈ X cl are commensurably terminal in Π X [cf. Proposition 2.3, (ii), below]. On the other hand, if D ⊆ Π X is a quasi-section, def and we write E = C Π X (D) ⊆ Π X for the commensurator of D in Π X [cf. §0], then one verifies immediately E is also a quasi-section. [Indeed, by considering the projection Π X  G k X , it follows immediately that every element of E centralizes some open subgroup D  ⊆ D; on the other hand, by considering the well-known properties of the action of open subgroups of G k on abelianizations of open subgroups of Δ X [i.e., more precisely, the “Riemann hypothesis for abelian varieties over finite fields” — cf., e.g., [Mumf], p. 206], it follows that every centralizer of D  in Δ X is trivial, i.e., that E Δ X = {1}.] (ii) It is immediate that any maximal subdecomposition group of Π X is, in fact, a decomposition group of some point ∈ X cl . On the other hand, since X is not necessarily Σ-separated, it is not clear whether or not every decomposition group of a point ∈ X cl is necessarily a maximal subdecomposition group. If X, Y are Σ-separated, then the arguments of [Tama], Corollary 2.10, Proposition 3.8, yield a “group-theoretic” characterization of the subdecomposition groups [hence also of the maximal subdecomposition groups, i.e., the decomposition groups of points ∈ X cl ] of Π X , Π Y in terms of the actions of the Frobenius elements. That is to say, if X, Y are Σ-separated, then any Frobenius-preserving isomorphism α is [quasi-]point-theoretic. (iii) Nevertheless, as was pointed out to the author by A. Tamagawa, even if X, Y are not necessarily Σ-separated, it is still possible to conclude, essentially from the arguments of [Tama], Corollary 2.10, Proposition 3.8, that: Any Frobenius-preserving isomorphism α is quasi-point-theoretic. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 33 Indeed, it suffices to give a “group-theoretic” characterization of the quasi- sections D ⊆ Π X which are decomposition groups of points ∈ X cl . We may assume [for simplicity] without loss of generality that X, Y are proper. Write def E = C Π X (D); k D , k E for the finite extension fields of k X determined by D, E. Let H ⊆ Δ X be a characteristic open subgroup; denote by Y → X the covering determined by the open subgroup E · H ⊆ Π X . Then it follows immediately from the definition of a “decomposition group” that it suffices to give a “group-theoretic” criterion for the condition that Y (k D ) contain a point whose field of definition [which is, a priori, some subextension in k D of k E ] is equal to k D . In [Tama], the Lefschetz trace formula is applied to compute the cardinality of Y (k D ). On the other hand, if we use the superscript “fld-def” to denote the subset of points whose field of definition is equal to the field given in parentheses, and “| − |” to denote the cardinality of a finite set, then for any subextension k  ⊆ k D of k E , we have  |Y (k  ) fld-def | |Y (k  )| = k  [where k  ⊆ k  ranges over the subextensions of k E ]. In particular, by apply- ing induction on [k  : k E ], one concludes immediately from the above formula that |Y (k  ) fld-def | may be computed from |Y (k  )| for subextensions k  ⊆ k  of k E [while |Y (k  )| may be computed, as in [Tama], from the Lefschetz trace formula]. This yields the desired “group-theoretic” characterization of the de- composition groups of Π X . Remark 11. Note that in the finite field case, if α as in Theorem 1.1, (i), (ii), is Frobenius-preserving, then the cardinalities, hence also the char- acteristics, of k X , k Y coincide. Indeed, this follows immediately by reducing to the proper case via Theorem 1.1, (i), and considering the actions of G k X , G k Y [cf. Theorem 1.1, (ii)] on M X , M Y [which are compatible relative to the ∼ isomorphism M X → M Y induced by α]. Now we return to the notation of the discussion preceding Theorem 1.1. Observe that the automorphism τ : X × X → X × X given by switching the two factors induces an outer automorphism of Π U X×X . Moreover, by choosing the basepoints used to form the various fundamental groups involved in an appropriate fashion, it follows that there exists an auto- morphism Π τ : Π U X×X → Π U X×X among those automorphisms induced by τ [i.e., all of which are related to one another by composition with an inner automorphism] which induces the auto- morphism on Π X×X = Π X × G k Π X given by switching the two factors; preserves 34 Shinichi Mochizuki the subgroup D X ⊆ Π U X×X ; and preserves and induces the identity automor- phism on the subgroup I X ⊆ D X (⊆ Π U X×X ). Note that by the slimness of Proposition 1.6, (i), together with the well-known commensurable terminality of D X ⊆ Π U X×X in Π U X×X [cf., e.g., [the proof of] [Mzk5], Lemma 1.3.12], it follows that, at least when Σ = Primes, these three conditions [are more than sufficient to] uniquely determine Π τ , up to composition with an inner au- tomorphism arising from I X ; one then obtains a natural Π τ for arbitrary Σ [well-defined up to composition with an inner automorphism arising from I X ] by taking the automorphism induced on the appropriate quotients by “Π τ in the case Σ = Primes”. Proposition 1.11. phism (Switching the Two Factors) The automor- c-ab : Π c-ab Π c-ab τ U X×X → Π U X×X induced by Π τ is the unique automorphism of the profinite group Π c-ab U X×X , up to composition with a cuspidally inner automorphism, that satisfies the following two conditions: (a) it preserves the quotient Π c-ab U X×X  Π X×X and induces on this quotient the automorphism on Π X×X = Π X × G k Π X given by switching the two factors; (b) it preserves the image of I X ⊆ D X → Π c-ab U X×X . Proof. This follows immediately from Proposition 1.10, (i). 2. Points and Functions We maintain the notation of §1 [i.e., the discussion preceding Theorem 1.1]. If x ∈ X cl , then we shall denote by D x ⊆ Π X the decomposition group of x [well-defined up to conjugation in Π X ]. If x ∈ X(k), then D x determines a section s x : G k → Π X [which is well-defined as a geometrically outer homomorphism]. Next, let S ⊆ X cl be a finite set. If n is a Σ † -integer [cf. §0], then the Kummer exact sequence 1 → μ n → G m → G m → 1 [where G m → G m is the n-th power map; μ n is defined so as to make the sequence exact] on the étale site of X determines a homomorphism Pic(X) → H 2 (Δ X , μ n ) [where Pic(X) is the Picard group of X]. Now there is a unique isomorphism ∼ μ n → M X /n · M X Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 35 such that the homomorphism Pic(X) → H 2 (Δ X , μ n ) sends line bundles of degree 1 to the element determined by 1 ∈ Z/nZ via the composite of the ∼ induced isomorphism H 2 (Δ X , μ n ) → H 2 (Δ X , M X /n·M X ) with the tautological ∼ 2 isomorphism H (Δ X , M X /n · M X ) → Z/nZ [cf. Proposition 1.2, (i)]. In the following discussion, we shall identify μ n with M X /n·M X via this isomorphism. If we consider the Kummer exact sequence on the étale site of U S ⊆ X [and pass to the inverse limit with respect to n], then we obtain a natural homomorphism × Γ(U S , O U ) → H 1 (Π U S , M X ) S [where we note that here, it suffices to consider the group cohomology of Π U S [i.e., as opposed to the étale cohomology of U S ], since the extraction of n-th × ) yields finite étale coverings of U S that corre- roots of an element of Γ(U S , O U S spond to open subgroups of Π U S ] which is injective [since the abelian topolog- × ) is clearly topologically finitely generated and free of p † - ical group Γ(U S , O U S torsion, hence injects into its prime-to-p † completion] whenever Σ † = Primes † . In particular, by allowing S to vary, we obtain a natural homomorphism × → lim H 1 (Π U S , M X ) K X −→ S [where K X is the function field of X; the direct limit is over all finite subsets S of X cl ] which is injective whenever Σ † = Primes † . Proposition 2.1. a finite subset, write (Kummer Classes of Functions) If S ⊆ X cl is c-cn Δ U S  Δ c-ab U S  Δ U S for the maximal cuspidally abelian and maximal cuspidally central quo- tients, respectively, and c-cn Π U S  Π c-ab U S  Π U S for the corresponding quotients of Π U S . If x ∈ X cl , then let us write D x [U S ] ⊆ Π U S for the decomposition group of x in Π U S [which is well-defined up to con- jugation in Π U S ] and I x [U S ] ⊆ D x [U S ] for the inertia subgroup. [Thus, when x ∈ S, we obtain [cf. Proposition 1.5, (ii), (iii)] a natural isomorphism of M X def with I x [U S ] = D x [U S ] Δ U S .] (i) The natural surjections induce isomorphisms as follows: ∼ ∼ 1 c-ab 1 H 1 (Π c-cn U S , M X ) → H (Π U S , M X ) → H (Π U S , M X ) 36 Shinichi Mochizuki In particular, we obtain natural homomorphisms as follows: ∼ ∼ × 1 c-ab 1 Γ(U S , O U ) → H 1 (Π c-cn U S , M X ) → H (Π U S , M X ) → H (Π U S , M X ) S ∼ ∼ × → lim H 1 (Π c-cn H 1 (Π c-ab H 1 (Π U S , M X ) K X U S , M X ) → lim U S , M X ) → lim −→ −→ −→ S S S These natural homomorphisms are injective whenever Σ = Primes † . † (ii) Suppose that S ⊆ X(k) is a finite subset. Then restricting cohomology classes of Π U S to the various I x [U S ], for x ∈ S, yields a natural exact sequence   ∧  † 1 → (k × ) → H 1 (Π U S , M X ) → Z x∈S  † ]. Moreover, the image [via the [where we identify Hom Z † (I x [U S ], M X ) with Z ∧ × natural homomorphism given in (i)] of Γ(U S , O U ) in H 1 (Π U S , M X )/(k × ) is S ∧ equal to the inverse image in H 1 (Π U S , M X )/(k × ) of the submodule of      † Z ⊆ Z x∈S x∈S determined by the principal divisors [with support in S]. A similar statement c-cn holds when “Π U S ” is replaced by “Π c-ab U S ” or “Π U S ”. × (iii) If f ∈ Γ(U S , O U ), write S κ c-cn ∈ H 1 (Π c-cn f U S , M X ); κ c-ab ∈ H 1 (Π c-ab f U S , M X ); κ f ∈ H 1 (Π U S , M X ) for the associated Kummer classes. If x ∈ X cl \S, then D x [U S ] maps, via the natural surjection Π U S  G k , isomorphically onto the open subgroup G k(x) ⊆ G k [where k(x) is the residue field of X at x]. Moreover, the images of the pulled back classes ∼ κ c-cn | D x [U S ] = κ c-ab | D x [U S ] = κ f | D x [U S ] ∈ H 1 (D x [U S ], M X ) → H 1 (G k(x) , M X ) f f ∼ ∧ → (k(x) × ) ∧ ∧ in (k(x) × ) are equal to the image in (k(x) × ) of the value of f at x. Proof. Assertion (i) follows immediately from the definitions. The exact sequence of assertion (ii) follows immediately from Proposition 1.4, (ii). The × ) is immediate from the definitions characterization of the image of Γ(U S , O U S and the exact sequence of assertion (ii). Assertion (iii) follows immediately from the definitions and the functoriality of the Kummer class. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 37 Remark 12. If, in the situation of Proposition 2.1, (iii), we think of the of Π extension of Π c-cn X as being given by the extension D S [cf. Proposition U S 1.6, (iii)], where D is a fundamental extension of Π X×X that appears as a ∧ quotient of Π U X×X [hence is “rigid” with respect to the action of (k × ) — cf. Proposition 1.7, (iii); the proof of Theorem 1.1, (iii)], then it follows that the image of D x [U S ] in Π c-cn U S may be thought of as the image of D x [U S ] in D S . If, moreover, we assume, for simplicity, that x ∈ X(k), S ⊆ X(k), then this image of D x [U S ] in D S amounts to a section of D S  Π X  G k lying over the section s x of Π X  G k . Since D S is defined as a certain fiber product, this section is equivalent to a collection of sections [regarded as cyclotomically outer homomorphisms] γ y,x : G k → D y,x [where y ranges over the points of S]. [Here, we note that it is immediate from the definitions that, as the notation suggests, γ y,x depends only on x, y — i.e., that γ y,x is independent of the choice of S.] That is to say, from this point of view, Proposition 2.1, (iii), may be regarded as stating that: ∧ ∧ × The image in (k × ) = (k(x) × ) of the value of a function ∈ Γ(U S , O U ) S at x ∈ X(k) may be computed from its Kummer class, as soon as one knows the sections γ y,x : G k → D y,x , for y ∈ S. Also, before proceeding, we note that an arbitrary section of D y,x  G k differs [as a cyclotomically outer homomorphism] from γ y,x by the action of an element ∼ ∧ of H 1 (G k , M X ) → (k × ) . Thus, the datum of “γ y,x ” may be regarded as a ∧ trivialization of a certain (k × ) -torsor. Remark 13. The finite field portion of Proposition 2.1 may be re- garded as the evident finite field analogue of [a certain portion of] the theory of [Mzk8], §4. Also, we observe that the approach of “reconstructing the func- tion field of the curve via Kummer theory, as opposed to class field theory [as was done in [Tama], [Uchi]]” has the advantage of being applicable to nonar- chimedean local fields, as well as to finite fields. Definition 2.1. For x, y ∈ X(k), we shall refer to the section [regarded as a cyclotomically outer homomorphism] γ y,x : G k → D y,x as the Green’s trivialization of D at (y, x). If D is a divisor on X supported in the subset of k-rational points X(k) ⊆ X cl , then multiplication of the various Green’s trivializations for the points in the support of D determines a section [regarded as a cyclotomically outer homomorphism] γ D,x : G k → D D,x 38 Shinichi Mochizuki which we shall refer to as the Green’s trivialization of D at (D, x). [Note that the definition of γ D,x generalizes immediately to the case where the divisor D, but not necessarily the points in its support, is rational over k — cf. Remark 5.] Remark 14. The terminology of Definition 2.1, is intended to sug- gest the similarity between the γ y,x of the present discussion and the “Green’s functions” that occur in the theory of bipermissible metrics — cf., e.g., [MB], §4.11.4. Remark 15. Note that the Green’s trivializations are symmetric with of Proposi- respect to the involution of D induced by the automorphism Π c-ab τ tion 1.11. Indeed, relative to the natural projections Π U X×X  Π c-ab U X×X  D the Green’s trivialization at (y, x) is simply the section of D  G k arising [by composition] from the section of Π U X×X  G k determined by the decomposi- tion group of the point (y, x) ∈ U X×X (k). Thus, the asserted symmetry of the is compatible with Π τ , Green’s trivializations follows from the fact that Π c-ab τ together with the evident fact that [by “transport of structure”] Π τ maps the decomposition group of (y, x) ∈ U X×X (k) isomorphically onto the decomposi- tion group of (x, y) ∈ U X×X (k). If d ∈ Z, denote by J d the subscheme of the Picard scheme of X that def parame-trizes line bundles of degree d; write J = J 0 . Thus, J d is a torsor over J. Note that there is a natural morphism X → J 1 [given by assigning to a point of X the line bundle of degree 1 determined by the point]. Thus, the basepoint of X [already chosen in §1] determines a basepoint of J 1 . At the level of “geometrically pro-Σ” étale fundamental groups, this morphism induces a surjective homomorphism Π X  Π J 1 whose kernel is the kernel of the maximal abelian quotient Δ X  Δ ab X . In particular, for x ∈ X(k), the section s x determines a section t x : G k → Π J 1 . Note that applying the “change of structure group” given by the “multiplication by d map” on J to the J-torsor J 1 yields the J-torsor J d . [Indeed, this follows by considering the group structure of the Picard scheme.] Thus, we obtain a morphism J 1 → J d whose induced morphism on fundamental groups Π J 1 → Π J d Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 39 determines an isomorphism of Π J d with the push-forward of the extension Π J 1 ab ab [i.e., of G k by Δ J 1 ∼ = Δ ab X ] via the homomorphism Δ X → Δ X given by mul- tiplication by d. When d ≥ 1, the group structure on the Picard scheme also determines a morphism  Π J 1 → Π J d [where the product is a fiber product over G k of d factors of Π J 1 ] which de- con- termines an isomorphism of Π J d with the push-forward  of the extension ab → Δ [i.e., from stituted by the fiber product via the homomorphism Δ ab X X ab a product of d copies of Δ ab X to Δ X given by adding up the d components]. Moreover, one verifies immediately that when d ≥ 1, these two constructions of “Π J d ” from Π J 1 yield groups that are naturally isomorphic. Thus, by applying the various homomorphisms induced on fundamental groups by the group structure of the Picard scheme, it follows that if D is any divisor of degree d on X whose support lies in the set of k-rational points X(k) ⊆ X cl , then D determines a section t D : G k → Π J d which may be constructed entirely group-theoretically from the “t x ”, where x ∈ X(k) ranges over the points in the support of D. In particular, if D is of degree 0, then the section t D : G k → Π J may be compared with the identity section of Π J to obtain a cohomology class: η D ∈ H 1 (G k , Δ ab X ) Now we have the following well-known result: Proposition 2.2. (Points and Galois Sections) Suppose that Σ = Primes. Then, in the notation of the above discussion: (i) The divisor D is principal if and only if η D = 0. (ii) The map x → D x from X cl to conjugacy classes of closed subgroups of Π X is injective, i.e., X is Primes-separated. Proof. First, we consider assertion (i). By well-known general nonsense [cf., e.g., [Naka], Claim (2.2); [NTs], Lemma (4.14); [Mzk4], the Remark pre- ceding Definition 6.2], there is a natural isomorphism ∼ ∧ H 1 (k, Δ ab X ) → J(k) (⊇ J(k)) [where the “∧” denotes the profinite completion] which maps η D to the ele- ment of J(k) determined by D. [Here, we recall that this natural isomorphism 40 Shinichi Mochizuki arises by considering the long exact sequence obtained by applying the functors H ∗ (G k , −) to the short exact sequence of G k -modules 1 → J(k)[n] → J(k) → J(k) → 1 — where n is a positive integer; the morphism J(k) → J(k) is the “multipli- cation by n map”; J(k)[n] is defined so as to make the sequence exact.] Thus, assertion (i) follows immediately. To prove assertion (ii), it suffices [by possibly base-changing to a finite ex- tension of k] to verify that two points x 1 , x 2 ∈ X(k) that induce Δ X -conjugate sections s x 1 , s x 2 are necessarily equal [cf. also [Tama], Corollary 2.10]. But this follows formally from assertion (i), by considering the divisor x 1 − x 2 [and the well-known fact that the natural morphism X → J 1 considered above is an embedding]. Remark 16. From the point of view of Definition 1.2, (ii), the reader may feel tempted to expect that [still under the assumption that Σ = Primes] D is principal if and only if the extension D D of Π X [by M X ] is trivial [i.e., determines the zero class in H 2 (Π X , M X )]. When k is nonarchimedean local, it is not difficult to verify, using Proposition 2.2, (i), that this is indeed the case. On the other hand, when k is finite, although this condition for principality is easily verified to be necessary, it is not, however, sufficient, since it only involves the “prime-to-p † portion” of the point of J(k) determined by D. Definition 2.2. In the situation of Theorem 1.1, (iii), suppose further def that (Σ = ) Σ X = Σ Y , and that α is point-theoretic. Let S ⊆ X cl be a [not ∼ necessarily finite] subset that corresponds via the bijection X cl → Y cl induced cl by [the point-theoreticity of] α to a subset T ⊆ Y . (i) Write D (respectively, E) for the fundamental extension of Π X×X (respec- c-ab tively, Π Y ×Y ) that arises as the quotient of Π c-ab U X×X (respectively, Π U Y ×Y ) by c-ab c-cn the kernel of the maximal cuspidally central quotient Δ U X×X  Δ U X×X (re- c-cn c-ab induces spectively, Δ c-ab U Y ×Y  Δ U Y ×Y ) [cf. Proposition 1.6, (iv)]. Thus, α an isomorphism: ∼ α c-cn : D → E We shall say that α is (S, T )-locally Green-compatible if, for every pair of points (x 1 , x 2 ) ∈ X(k X )×X(k X ) corresponding via the bijection induced by α to a pair of points (y 1 , y 2 ) ∈ Y (k Y ) × Y (k Y ), such that x 2 ∈ S, y 2 ∈ T , the isomorphism ∼ D x 1 ,x 2 → E y 1 ,y 2 [obtained by restricting α c-cn ] is compatible with the Green’s trivializations. We shall say that α is (S, T )-locally degree zero (respectively, (S, T )-locally Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 41 principally) Green-compatible if, for every x ∈ X(k X ) S and every divisor of degree zero (respectively, principal divisor) D supported in X(k X ) ⊆ X cl corresponding via the bijection induced by α to a pair (y, E) of Y [so y ∈ Y (k Y ) T ], the isomorphism ∼ D D,x → E E,y is compatible with the Green’s trivializations. (ii) We shall say that α is totally (S, T )-locally Green-compatible (respectively, totally (S, T )-locally degree zero Green-compatible; totally (S, T )-locally prin- cipally Green-compatible) if, for all pairs of connected finite étale coverings X  → X, Y  → Y that arise from open subgroups of Π X , Π Y that correspond via α, the isomorphism ∼ Π X  → Π Y  induced by α is (S  , T  )-locally Green-compatible (respectively, (S  , T  )-locally degree zero Green-compatible; (S  , T  )-locally principally Green-compatible), where S  ⊆ (X  ) cl , T  ⊆ (Y  ) cl are the inverse images in X  , Y  of S, T , respectively. (iii) With respect to the terminology introduced in (i), (ii), when S = X cl , T = Y cl , then we shall replace the phrase “(S, T )-locally” by the phrase “globally”. Remark 17. In the situation of Definition 2.2, if X  → X, Y  → Y are connected finite étale coverings that arise from open subgroups of Π X , Π Y ∼ that correspond via α; D → E is the isomorphism of fundamental extensions of Π X×X , Π Y ×Y that arises from the isomorphism α c-ab of Theorem 1.1, (iii); and the points x 1 , x 2 (respectively, y 1 , y 2 ) are Δ X - (respectively, Δ Y -) conjugate, then it follows immediately from the compatibility of α c-ab with the natural c-ab inclusions D X → Π c-ab U X×X , D Y → Π U Y ×Y [cf. Theorem 1.1, (iii)] that the ∼ isomorphism D x 1 ,x 2 → E y 1 ,y 2 is automatically compatible with the Green’s triv- ializations. [Indeed, this follows from the easily verified fact that the Green’s trivializations in this case are, in essence, specializations of conjugates of the  ” of Proposition 1.8.] Unfortunately, however, the au- “canonical sections of ζ = thor is unable, at the time of writing, to see how to generalize the argument applied in the proof of Theorem 1.1, (iii), involving Lemma 1.1; Proposition 1.8, (v), so as to cover the case where the points x 1 , x 2 (respectively, y 1 , y 2 ) fail to be Δ X - (respectively, Δ Y -) conjugate. Remark 18. It is immediate that (S, T )-local Green-compatibility (re- spectively, (S, T )-local degree zero Green-compatibility) implies (S, T )-local de- gree zero Green-compatibility (respectively, (S, T )-local principal Green-compa- tibility), and that total (S, T )-local Green-compatibility (respectively, total 42 Shinichi Mochizuki (S, T )-local degree zero Green-compatibility) implies total (S, T )-local degree zero Green-compatibility (respectively, total (S, T )-local principal Green-compa- tibility). Theorem 2.1. (Reconstruction of Functions) In the situation of def Theorem 1.1, (iii), suppose further that (Σ = ) Σ X = Σ Y , and that α is point- theoretic. Then: (i) Let S ⊆ X cl , T ⊆ Y cl be finite subsets that correspond via the bijection ∼ X cl → Y cl induced by α. Then α, α c-ab induce isomorphisms [well-defined up to cuspidally inner automorphisms] ∼ c-ab Π c-ab U S → Π V T def [where V T = Y \T ] lying over α, which are functorial with respect to α and S, T , as well as with respect to passing to connected finite étale coverings of X, Y [that do not necesarily arise from open subgroups of Π X , Π Y !]. ∼ (ii) Suppose that Σ = Primes. Then the bijection X cl → Y cl induced by α induces a bijection between the groups of principal divisors on X, Y . This bijection, together with the isomorphisms of (i), induces a compatible iso- morphism ∧ ∼ ∧ × × · (k X ) → K Y × · (k Y × ) K X between the push-forwards of the multiplicative groups associated to the func- × × ∧ → (k X ) , k Y × → tion fields of X, Y , relative to the homomorphisms k X ∧ (k Y × ) . Proof. Assertion (i) follows immediately by “specializing to S, T ” the iso- morphism of Theorem 1.1, (iii) [cf. also Proposition 1.9, (i), (ii); the definitions of the various objects involved]. [Here, we note that the functoriality asserted in assertion (i), which is somewhat stronger than the functoriality asserted in Theorem 1.1, (iii), follows from the definitions, together with the naturality of the constructions applied in the proof of Theorem 1.1, (iii) — cf., e.g., the dia- gram of Proposition 1.7, (ii).] Assertion (ii) follows immediately from assertion (i); Proposition 2.2, (i); Proposition 2.1, (i), (ii). ∼ c-ab Remark 19. In fact, the crucial isomorphism Π c-ab U S → Π V T of Theo- rem 2.1, (i), may also be constructed, in the finite field case, via the techniques to be introduced in §3 [although we shall not discuss this approach in detail; Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 43 cf., however, the proof of Theorem 3.1]. On the other hand, observe that unlike the techniques of §3, the techniques of §1 [in particular, the proof of Theorem 1.1, (iii), via Propositions 1.7, 1.8] apply to situations [e.g., the case of nonar- chimedean local fields!] where the weight filtration [cf. §3] does not admit a Galois-invariant splitting. Indeed, the techniques of §1, essentially only require that the Galois cohomology of the base field admit a natural duality pairing. Moreover, even in the finite field case, in light of the importance of this iso- ∼ c-ab morphism Π c-ab U S → Π V T in the theory of the present paper, it is of interest to see that this isomorphism may be constructed via two fundamentally dif- ferent approaches. Finally, although the techniques of §3 are better suited to the reconstruction of the Green’s trivializations, they have the drawback that they depend essentially on the choice of a “basepoint” x ∗ ∈ X(k). Thus, it is of interest to know that this isomorphism may be constructed [i.e., via the techniques of §1] “cohomologically” [cf. Proposition 1.5, (i)] without making such a choice. Remark 20. In the case of nonarchimedean local fields, it is natural to ask, in the style of [Mzk8], §4, whether or not various “canonical integral struc- tures” on the extensions D x,y [where x, y ∈ X(k)] of G k by M X are preserved by arbitrary isomorphisms of arithmetic fundamental groups. When x = y, such a canonical integral structure is determined by the Green’s trivialization; when x = y, such a canonical integral structure is determined by the integral structure [in the usual sense of scheme theory] on the canonical sheaf of the stable model of the curve [when the curve has stable reduction] — cf. [Mzk8], §4. Before proceeding, we note the following “analogue for Π c-ab U S ” of Proposi- tion 1.10, (i): Proposition 2.3. (Automorphisms and Commensurators) Let Π c-ab be as in Proposition 2.1. For x ∈ S, write D x [U S ] → Π c-ab U S U S for the natural inclusion. Then: (i) Any automorphism α of the profinite group Π c-ab U S which (a) is compatible with the natural surjection Π c-ab U S  Π X and induces the identity on Π X ; (b) for each x ∈ S, preserves the image of M X ∼ = I x [U S ] ⊆ D x [U S ] via the natural inclusion D x [U S ] → Π c-ab U S 44 Shinichi Mochizuki is cuspidally inner. (ii) Suppose that X is Σ-separated. Then for x ∈ S, D x is commensurably terminal in Π X . (iii) Suppose that X is Σ-separated. Then the image of D x [U S ] → Π c-ab U S is commensurably terminal in Π c-ab U S . Proof. First, we observe that assertion (ii) follows formally from the def- inition of a “decomposition group” and “Σ-separated”. Thus, assertion (i) (respectively, (iii)) follows by an argument which is entirely similar to the ar- gument that was used to prove assertion (i) (respectively, (iii)) of Proposition 1.10. Remark 21. In the situation of Definition 2.2, suppose that S, T are finite, and that α arises from an isomorphism ∼ Π U S → Π V T which is point-theoretic [or, equivalently, quasi-point-theoretic] — a condition that is automatically satisfied in the finite field case whenever α is Frobenius- preserving [cf. Remark 10]. Then observe that, [in light of our point-theoreticity assumption] it follows from Proposition 2.3, (i), that the resulting induced isomorphism ∼ c-ab Π c-ab U S → Π V T coincides [up to cuspidally inner automorphisms] with the isomorphism of Theo- rem 2.1, (i). Thus, in light of Remark 15, it follows formally from the definitions that α is totally (S, T )-locally Green-compatible. Corollary 2.1. (Point-theoretic Totally Locally Principally Green- compatible Isomorphisms) In the situation of Theorem 1.1, (iii), assume def further that (Σ = ) Σ X = Σ Y = Primes, and that α is point-theoretic and totally (S, T )-locally principally Green-compatible, for some nonempty ∼ subsets S ⊆ X cl , T ⊆ Y cl which correspond via the bijection X cl → Y cl in- duced by α. Then α arises from a uniquely determined commutative diagram of schemes ∼  → X Y  ⏐ ⏐ ⏐ ⏐ X ∼ → Y Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 45 in which the horizontal arrows are isomorphisms; the vertical arrows are the pro-finite étale coverings determined by the profinite groups Π X , Π Y . Proof. Corollary 2.1 follows immediately — i.e., by “specializing func- tions to points” — from the definitions; Theorem 2.1, (ii); Proposition 2.1, (iii); Remark 12; and [Tama], Lemma 4.7. Here, we note that, in the present situation, the isomorphism ∧ ∼ ∧ × × · (k X ) → K Y × · (k Y × ) K X ∼ × of Theorem 2.1, (ii), necessarily induces an isomorphism K X → K Y × [cf. the assumption that Σ † = Primes † ]. Indeed, this is immediate in the finite field case. In the nonarchimedean local field case, it follows via the arguments applied in the proof of [Mzk8], Theorem 4.10: That is to say, we assume for simplicity × , and x ∈ S is a point that does not lie in the that S ⊆ X(k X ); then if f ∈ K X divisor of zeroes and poles of f , then let us observe that the subset ∧ × × ⊆ f · (k X ) f · k X may be characterized as the subset of elements whose values [cf. Proposition × × ∧ 2.1, (iii)] at x lie in k X ⊆ (k X ) . Note that since, for a given x 1 ∈ S, there × clearly exist f ∈ K X [at least after possibly passing to an appropriate connected finite étale covering of X] that have a zero or pole at x 1 but not at some other x ∈ S, this observation allows us to recover the canonical discrete structure [cf. [Mzk8], Definition 4.1, (iii); the proof of [Mzk8], Theorem 4.10] on the cl is an arbitrary finite subset decomposition groups in Π c-ab U S 1 [where S 1 ⊆ X containing S, which corresponds, say, to a subset T 1 ⊆ Y cl that contains T ] at arbitrary points [i.e., arbitrary “x 1 ”] of S. Thus, by applying this canonical discrete structure [as in the proof of [Mzk8], Theorem 4.10], we may recover the subset ∧ × × f · k X ⊆ f · (k X ) × for arbitrary f ∈ K X [i.e., even f that have a zero or pole at every point of S] as the subset of elements for which the restriction to each point x of S either × × ∧ ⊆ (k X ) or [when the element in question has a zero or pole at x] lies in k X is compatible with the canonical discrete structure at x. Since this characteri- × × ∧ ⊆ f · (k X ) is manifestly compatible [in light of the zation of the subset f · k X ∼ c-ab Green-compatibility assumption on α] with the isomorphisms Π c-ab U S 1 → Π V T 1 induced by α, we thus conclude that the isomorphism ∧ ∼ ∧ × × · (k X ) → K Y × · (k Y × ) K X ∧ × × × of Theorem 2.1, (ii), maps the subset K X ⊆ K X · (k X ) onto the subset K Y × ⊆ ∧ K Y × · (k Y × ) , as desired. 46 Shinichi Mochizuki Remark 22. Suppose, in the situation of Corollary 2.1, that S = X cl , cl T = Y . Then unlike the situation discussed in [Tama], one has the freedom to evaluate functions at arbitrary points of the entire sets X cl , Y cl , as opposed to just certain restricted subsets S ⊆ X cl , T ⊆ Y cl . Thus, instead of applying [Tama], Lemma 4.7, one may instead apply the somewhat easier argument implicit in [Uchi], §3, Lemmas 8-11 [which is used to treat the function field case]. Thus, in light of Remark 21 [together with the portion of Theorem 1.1, (i), concerning the preservation of decomposition groups of cusps], Corollary 2.1 implies the following result, in the affine case: Corollary 2.2. (Point-theoretic Isomorphisms in the Affine Case) Let U , V be affine hyperbolic curves over a finite or nonarchimedean local field. Suppose that Σ = Primes. Write Δ U (respectively, Δ V ) for the maximal cuspidally pro-Σ † quotient of the maximal pro-Σ quotient of the tame geometric fundamental group of U (respectively, V ) [where “tame” is with respect to the complement of U (respectively, V ) in its canonical com- pactification], and Π U (respectively, Π V ) for the corresponding quotient of the étale fundamental group of U (respectively, V ). Then any point-theoretic isomorphism ∼ β : Π U → Π V arises from a uniquely determined commutative diagram of schemes  U ∼ → ⏐ ⏐ U V  ⏐ ⏐ ∼ → V in which the horizontal arrows are isomorphisms; the vertical arrows are the pro-finite étale coverings determined by the profinite groups Π U , Π V . Remark 23. In light of the results of [Tama] [cf. Remarks 9, 10], Corollary 2.2 is only truly of interest in the case of nonarchimedean local fields. Definition 2.3. Suppose that k is a nonarchimedean local field. (i) A [necessarily affine] hyperbolic curve U over k will be said to be of strictly Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 47 Belyi type if it is defined over a number field and isogenous [cf. §0] to a hyper- bolic curve of genus zero. (ii) A [necessarily affine] hyperbolic curve U over k will be said to be of Belyi type if it is defined over a number field, and, moreover, for some positive integer m, there exists a finite sequence U = U 1  U 2  . . .  U m−1  U m of hyperbolic orbicurves [cf. §0] U j such that U m is a tripod [cf. §0], and, moreover, for each j = 1, . . . , m−1, U j+1 is related to U j in one of the following ways: (a) there exists a finite étale morphism U j+1 → U j [i.e., “U j+1 is a finite étale covering of U j ”]; (b) there exists a finite étale morphism U j → U j+1 [i.e., “U j+1 is a finite étale quotient of U j ”]; (c) there exists an open immersion U j → U j+1 [i.e., in the terminology of [Mzk8], “U j+1 is a [hyperbolic] partial compactification of U j ”]; (d) there exists a partial coarsification morphism [cf. §0] U j → U j+1 [i.e., “U j+1 is a partial coarsification of U j ”]. (iii) A [necessarily affine] hyperbolic curve U over k will be said to be of quasi- Belyi type if it is defined over a number field and admits a connected finite étale covering V → U such that V admits a [not necessarily finite or étale!] dominant morphism V → W to a tripod W . Remark 24. It is immediate that every hyperbolic curve of strictly Belyi type is also of Belyi type [as the terminology suggests]. Moreover, one verifies easily by “induction on m” [where “m” is as in Definition 2.3, (ii)] that every hyperbolic curve of Belyi type is also of quasi-Belyi type [as the termi- nology suggests]. It is not difficult to see that there exist [multiply] punctured elliptic curves that are of Belyi type, but not of strictly Belyi type [cf. Remark 31 below]. On the other hand, it is not clear to the author at the time of writing whether or not there exist hyperbolic curves of quasi-Belyi type that are not of Belyi type. Remark 25. Hyperbolic curves of strictly Belyi type are precisely the sort of curves considered in [Mzk8], Corollaries 2.8, 3.2. 48 Shinichi Mochizuki Remark 26. The author would like to thank A. Tamagawa for useful discussions concerning Definition 2.3, (ii), especially Definition 2.3, (ii), (d). Proposition 2.4. (Decomposition Groups of Curves of Quasi- Belyi Type) Let U (respectively, V ) be a hyperbolic curve over a nonar- chimedean local field. Denote the base field of U (respectively, V ) by k U (re- spectively, k V ), the étale fundamental group of U (respectively, V ) by Π U (re- spectively, Π V ) [i.e., “we take Σ = Primes”]. Let ∼ β : Π U → Π V be an isomorphism of profinite groups. Then: (i) If U is of quasi-Belyi type, then the closed points of “DLoc-type” [in the sense of [Mzk8], Definition 2.4] are p U -adically dense [where p U is the residue characteristic of k U ] in U (k U ). (ii) If U is of quasi-Belyi type, then β maps every decomposition group of a closed point of U isomorphically onto a decomposition group of a closed point of V . (iii) If both U , V are of quasi-Belyi type, then β is point-theoretic. (iv) If U is of Belyi type, then so is V . Proof. The proof of assertion (i) is similar to the proof of [Mzk8], Corol- lary 2.8: That is to say, in the terminology of loc. cit., it follows formally from the fact that U is of quasi-Belyi type that the “algebraic” closed points [i.e., closed points defined over a number field, which are manifestly p U -adically dense in U (k U )] of U are of “DLoc-type” [cf. the proof of [Mzk8], Corollary 2.8]: Indeed, it suffices to consider the following commutative diagram of hyperbolic curves, whose existence follows from the assumption that U is of quasi-Belyi type: U ←− V  −→ W  ⏐ ⏐ ⏐ ⏐ V −→ → U  −→ U W Here, the “hooked arrow →” is an open immersion; all of the “non-hooked Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 49 arrows” except for V → W , V  → W  are finite étale morphisms; V → W , V  → W  are dominant; the finite étale morphism U  → U is obtained by a base- change to a finite extension of the base field k U ; and W is a tripod [so W  → W is a “Belyi map”]. Note that the composite arrow V  → W  → U  → U may be thought of as an arrow in the category DLoc k U (U ) of [Mzk8], §2. Observe, moreover, that the arrow W  → U  may be chosen to have arbitrarily designated algebraic closed points in the complement of its image. Thus, we conclude that this diagram exhibits the [arbitrarily designated] algebraic closed points in the complement of the image of W  → U  → U as points of DLoc-type, as desired. This completes the proof of assertion (i). In light of assertion (i) [applied to the various connected finite étale cover- ings of U ], the proof of assertion (ii) is entirely similar to the proof of [Mzk8], Corollary 3.2: That is to say, by [Mzk8], Corollary 2.5, it follows that β maps decomposition groups of DLoc-type of U to decomposition groups of DLoc-type of V . Thus, assertion (ii) follows by applying [Mzk8], Lemma 3.1 [where the density statement of assertion (i) concerning points of DLoc-type allows one to replace the “algebraicity” condition of [Mzk8], Lemma 3.1, (iii), by the condi- tion that the points in question be of DLoc-type]. Finally, assertion (iii) follows formally from assertion (ii) [and Proposition 2.2, (ii)]. Finally, we consider assertion (iv). First, I claim that by applying the isomorphism β [and thinking of hyperbolic orbicurves as being represented by their associated étale fundamental groups], one may transform the sequence U = U 1  U 2  . . .  U m−1  U m of Definition 2.3, (ii), into a sequence V = V 1  V 2  . . .  V m−1  V m that also satisfies the conditions of Definition 2.3, (ii), in such a way that we ∼ also obtain compatible isomorphisms β j : Π U j → Π V j [where j = 1, . . . , m; β 1 = β]. Indeed, we reason by induction on m. If [for j = 1, . . . , m − 1] U j+1 is related to U j as in (a) [of Definition 2.3, (ii)], then it is immediate [by thinking in terms of open subgroups of Π U j , Π V j ] that one may construct [from V j ] a V j+1 related to V j as in (a). If U j+1 is related to U j as in (b) (respectively, (c)), then it follows from [Mzk6], Theorem 2.4 (respectively, [Mzk8], Theorem 1.3, (iii) [cf. also [Mzk8], Theorem 2.3]), that one may construct [from V j ] a V j+1 related to V j as in (b) (respectively, (c)). If U j+1 is related to U j as in (d), then Π U j+1 is obtained from Π U j by forming the quotient of Π U j by the closed normal subgroup of Π U j generated by some finite collection of elements of Δ U j that belong to the decomposition groups of points of U j in Δ U j . Thus, by Lemma 2.1, (v), below, we conclude that the quotient Π U j  Π U j+1 determines a quotient Π V j  Π V j+1 that corresponds to a partial coarsification V j → V j+1 , as desired. ∼ Finally, if U m is a tripod, the existence of the isomorphism Π U m → Π V m implies that V m is also a tripod [cf. [Mzk5], Lemma 1.3.9]. This completes the proof of the claim. Thus, to complete the proof of assertion (iv), it suffices to verify that V is defined over a number field. But observe that since U is defined over a number 50 Shinichi Mochizuki field, there exists a diagram of hyperbolic curves [i.e., in essence, a “Belyi map”] U m ←−  U m → U  −→ U where the “hooked arrow →” is an open immersion; the “non-hooked arrows” are finite étale morphisms; and the finite étale morphism U  → U is obtained by a base-change to a finite extension of the base field k U . Now the isomorphisms ∼ ∼ Π U m → Π V m , Π U → Π V allow us to transform [cf. [Mzk8], Theorem 2.3 and its proof] this diagram into a similar diagram V m ←− V m  → V  −→ V whose existence [since V m is also a tripod!] shows that V is also defined over a number field, as desired. This completes the proof of assertion (iv). Remark 27. Note that the essential reason that the author is unable to prove the stronger statement of Proposition 2.4, (iv), in the quasi-Belyi case is that, in the notation of the proof of Proposition 2.4, (i), it is unclear how to construct [at the level of arithmetic fundamental groups] the dominant morphism V → W from V . That is to say, unlike the situation involving the operations of Definition 2.3, (ii), (a), (b), (c), (d), it is by no means clear how to construct, via purely group-theoretic operations, the quotient of an arithmetic fundamental group arising from an arbitrary dominant morphism. Lemma 2.1. (Finite Subgroups of Fundamental Groups of Hyperbolic Orbicurves) Let W be a hyperbolic orbicurve over an alge- braically closed field of characteristic zero; Σ W a nonempty set of prime num- bers. Denote the maximal pro-Σ W quotient of the étale fundamental group of W by Δ W ; suppose that W admits a finite étale covering by a hyperbolic curve that arises from an open subgroup of Δ W . Let A ⊆ Δ W (respectively, B ⊆ Δ W ) be the decomposition group [well-defined up to conjugation in Δ W ] of a closed point w A (respectively, w B ) of W ; suppose that w A = w B . Then: (i) A, B are cyclic. (ii) A Δ W . B = {1}. In particular, if A = {1}, then A is normally terminal in (iii) The order of every finite cyclic closed subgroup C ⊆ Δ W divides the order of W [cf. §0]. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 51 (iv) Every finite nontrivial closed subgroup C ⊆ Δ W is contained in a decomposition group of a unique closed point of W . (v) The nontrivial decomposition groups of closed points of W may be charac- terized as the maximal finite nontrivial closed subgroups of Δ W . Proof. Assertion (i) follows immediately from the well-known [and easily verified] fact that the absolute Galois group of a complete discrete valuation field with algebraically closed residue field of characteristic zero is cyclic. Next, we consider assertion (ii). Let C ⊆ A B be a subgroup of prime order l ∈ Σ W . Now consider a normal open subgroup H ⊆ Δ W such that the covering W H → W determined by H is a hyperbolic curve. Note that this implies that A H = B H = C H = {1} [cf., e.g., assertion (iii), which will be proven below without applying the present assertion (ii)]. Write W H → W C → W for the covering determined by the open subgroup C · H ⊆   , w B of W C that lift w A , w B , Δ W . Observe that there exist closed points w A respectively, and whose decomposition groups [well-defined up to conjugation in C · H] are equal to C. Note that since W H is a hyperbolic curve, and C is of prime order l, it follows that the order of every closed point of W C is equal to either 1 or l. Now if W C is affine, then let v be a cusp of W C . If W C is proper and admits ≥ 3 points of order l, then let v be a point of W C of order l such   , w B . Note that if W C is proper and admits ≤ 2 points of order l, that v = w A then it follows from the hyperbolicity assumption that the coarsification of W C is a proper smooth curve of genus ≥ 1; thus, by replacing H by an appropriate open subgroup of H, one verifies immediately that one may assume without loss of generality that either W C is affine or W C admits ≥ 3 points of order l. Now observe that W C admits a finite étale cyclic covering W C  → W C of degree l which is étale over the compactification of the coarsification of W C , except over the points in the compactification of the coarsification of W C corresponding  , over which W C  is totally ramified. In particular, it follows that any to v, w B   (respectively, w B ) is of order l (respectively, 1), thus point of W C  lying over w A contradicting the observation that the decomposition groups [well-defined up   , w B are equal to C. This completes the proof to conjugation in C · H] of w A that A B = {1}. By applying this fact to arbitrary finite étale coverings of W , it follows formally [cf. Proposition 2.3, (ii)] that A is normally terminal in Δ W , whenever A = {1}. Next, we consider assertion (iii). Denote the order of W by n. Now if C ⊆ Δ W is a nontrivial finite cyclic closed subgroup, then there exists a normal open subgroup N ⊆ Δ W such that C N = {1}. In particular, it follows that def if we take H = C · N [so H ⊆ Δ W is an open subgroup], then the natural map C → H ab is injective. On the other hand, if we denote by W H → W the covering determined by H, then it is clear that the order of W H divides n, hence that H ab is an extension of a torsion-free profinite abelian group by a finite abelian group annihilated by n. Thus, we conclude from the injection 52 Shinichi Mochizuki C → H ab that the order of C divides n, as desired. This completes the proof of assertion (iii). Next, we consider assertion (iv). First, let us observe that uniqueness follows formally from assertion (ii). Next, let us verify assertion (iv) under the further assumption that C is solvable. By induction on the order of C, we may assume that [at least] one of the following conditions is satisfied: (a) C is an extension of a group of prime order by a nontrivial subgroup C 1 ⊆ C which is contained in the decomposition group A; (b) C is of prime order l ∈ Σ W . If (a) is satisfied, then by replacing W by a finite étale covering of W determined by a suitable open subgroup containing C, we may assume that (C 1 ⊆) A ⊆ C. Thus, if A = C, then A = C 1 is normal in C. But this implies, by the normal terminality portion of assertion (ii), that A = C, a contradiction. Thus, (a) implies that C ⊆ A. If (b) is satisfied, then we argue as follows: Observe that by assertion (iii), every open subgroup H ⊆ Δ W that contains C determines a finite étale covering W H → W such that the order of W H is divisible by l. Write Stack l (W H ) for the set of closed points of W H whose order is divisible by l. Now observe that since the order of W H is divisible by the prime number l, it follows that Stack l (W H ) is nonempty. Since the set Stack l (W H ) is finite and nonempty, we thus conclude that, if we allow H to vary [among open subgroups H ⊆ Δ W that contain C], then the inverse limit Stack l (W H ) lim ←− H is nonempty. But, unraveling the definitions, this means precisely that C con- tains the decomposition group D associated to some compatible system of points of the sets Stack l (W H ). Since D is of order divisible by l, we thus conclude that D = C, as desired. This completes the proof of assertion (iv) for C solvable. On the other hand, a well-known theorem from the theory of finite groups asserts that a finite group in which every Sylow subgroup is cyclic is solvable [cf. [Scott], p. 356]. Thus, in light of assertion (i), we conclude that assertion (iv) for C solvable implies assertion (iv) for C arbitrary. Finally, we observe that assertion (v) follows formally from assertions (ii), (iv). Remark 28. The author would like to thank A. Tamagawa for inform- ing him of Lemma 2.1 and, in particular, of the theorem on finite groups that was applied in the proof of Lemma 2.1, (iv). We are now ready to state the following “absolute p-adic version of the Grothendieck Conjecture” for hyperbolic curves of Belyi or quasi-Belyi type: Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 53 Corollary 2.3. (Curves of Belyi or Quasi-Belyi Type) Let U (respectively, V ) be a hyperbolic curve over a nonarchimedean local field. Denote the base field of U (respectively, V ) by k U (respectively, k V ), the étale fundamental group of U (respectively, V ) by Π U (respectively, Π V ) [i.e., “we take Σ = Primes”]. Suppose further that at least one of the following condi- tions holds: (a) both U and V are of quasi-Belyi type; (b) either U or V [but not necessarily both!] is of Belyi type. Then any isomorphism of profinite groups ∼ β : Π U → Π V arises from a uniquely determined commutative diagram of schemes  U ∼ → ⏐ ⏐ ⏐ ⏐ U V  ∼ → V in which the horizontal arrows are isomorphisms; the vertical arrows are the pro-finite étale coverings determined by the profinite groups Π U , Π V . Proof. In light of Proposition 2.4, (iii), (iv) [cf. also Remark 24], Corollary 2.3 follows formally from Corollary 2.2. Remark 29. Note that in the proof of Proposition 2.4, Corollary 2.3, it is necessary, in the quasi-Belyi case, to apply the full “Hom version” of [Mzk4], Theorem A. This differs from the situation of [Mzk8], Corollaries 2.8, 3.2 — i.e., where one only treats hyperbolic curves of strictly Belyi type — or, indeed, of the portion of Proposition 2.4, Corollary 2.3, that concerns curves of Belyi type, in which the “isomorphism version” of [Mzk4], Theorem A, suffices [cf. [Mzk8], Remark 2.8.1]. Thus, in the terminology of [Mzk6], Definition 3.7, the portion of Corollary 2.3 concerning hyperbolic curves of Belyi type admits the following formal consequence: 54 Shinichi Mochizuki Corollary 2.4. (Absoluteness of Curves of Belyi Type) Every hyperbolic curve of Belyi type over a nonarchimedean local field is absolute. Remark 30. It is interesting to note that the essential property that underlies the absoluteness of Corollary 2.4 is the existence of a Belyi map [since the curve is defined over a number field], which, in the context of the theory of [Mzk8], §2, may be regarded as a sort of endomorphism of the curve. From this point of view, Corollary 2.4 is reminiscent of [Mzk6], Corollary 3.8, which states that the “canonical curves” of p-adic Teichmüller theory are absolute. Indeed, from the point of view of the theory of [Mzk2], this canonicality may be regarded as the existence of a sort of “Frobenius endomorphism” of the curve. It is also interesting to note that both of these results assert that every member of some countable collection of nonarchimedean hyperbolic curves is absolute. Remark 31. In the context of Remark 30, it is interesting to note that, unlike the canonical curves discussed in [Mzk6], §3, the set of points determined by the hyperbolic curves of strictly Belyi type fails, for all pairs (g, r) such that 2g − 2 + r ≥ 3, g ≥ 1, to be Zariski dense in the moduli stack of hyperbolic curves of type (g, r). Indeed, this follows immediately from [Mzk1], Theorem B. On the other hand, it is not clear to the author at the time of writing whether or not the set of points determined by the hyperbolic curves of Belyi (respectively, quasi-Belyi) type is Zariski dense in the moduli stack of hyperbolic curves of type (g, r) [when, say, 2g − 2 + r ≥ 3, g ≥ 2]. Note, however, that when g = 0, 1, [one verifies easily that] every hyperbolic curve of type (g, r) that is defined over a number field is automatically of Belyi type. 3. Maximal Pro-l Cuspidalizations In this §, we apply the theory of the weight filtration [cf. [Kane], [Mtm]], to- gether with various generalities concerning free Lie algebras [cf. the Appendix], to construct, in the finite field case, “maximal cuspidally pro-l cuspidalizations” [cf. Theorem 3.1], whose existence implies, under quite general conditions [cf. Corollary 3.1 below], that an isomorphism “α” as in Theorem 1.1, (iii), is always totally globally Green-compatible. In the following discussion, we maintain the notation of §2, and assume further throughout the present §3 that we are in the finite field case. Definition 3.1. Let l be a prime number; G, H, A topologically finitely generated pro-l groups; φ : H → A a [continuous] homomorphism. Suppose further that A is abelian, and that G is an l-adic Lie group [cf., e.g., [Serre], Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 55 Chapter V, §7, §9, for basic facts concerning l-adic Lie groups]. (i) We shall refer to as the φ-central filtration on H the filtration defined as follows: def H(1) = H def H(m) = def H(2) = Ker(φ)  the subgroup topologically generated by the commutators  [H(a), H(b)], where a + b = m, ∀ m ≥ 3 Thus, in words, this filtration on H is the “fastest decreasing central filtration among those central filtrations whose top quotient factors through φ”. We shall say that H is φ-nilpotent if H(m) = {1} for sufficiently large m. If H is φ-nilpotent when φ is taken to be the natural surjection H  H ab to its abelianization H ab , then we shall say that H is nilpotent. In the following, for a, b, n ∈ Z such that 1 ≤ a ≤ b, n ≥ 1, we shall write def H(a/b) = H(a)/H(b) and def Gr(H)(n) =  def H(m/m + 1) ⊆ Gr(H) = Gr(H)(1) m≥n def Gr(H)(a/b) = Gr(H)(a)/Gr(H)(b) and append a subscript Q l (respectively, F l ) to these objects to denote the result of tensoring over Z l with Q l (respectively, F l ). Thus, Gr(H), Gr Q l (H), Gr F l (H) are graded Lie algebras over Z l , Q l , F l , respectively; Gr(H)(n) ⊆ Gr(H) is a [Lie algebra-theoretic] ideal. Also, if Z a ≥ 1, then we shall write: def H(a/b) H(a/∞) = lim ←− b [where b ranges over the integers ≥ a + 1]. (ii) We shall denote by Lie(G) the Lie algebra over Q l determined by G. If G is nilpotent, then Lie(G) is a nilpotent Lie algebra over Q l , hence determines a connected, unipotent linear algebraic group Lin(G), which we shall refer to as the linear algebraic group associated to G. In this situation, there exists [cf., e.g., Remark 33 below] a natural [continuous] homomorphism [with open image] G → Lin(G)(Q l ) [from G to the l-adic Lie group determined by the Q l -valued points of Lin(G)] which is uniquely determined [since Lin(G) is connected and unipotent!] by the 56 Shinichi Mochizuki condition that it induce the identity morphism on the associated Lie algebras. In the situation of (i), if Z a ≥ 1, then we shall write: def Lie(H(a/b)); Lie(H(a/∞)) = lim ←− def Lin(H(a/∞)) = lim Lin(H(a/b)) ←− b b [where b ranges over the integers ≥ a + 1; we recall that it is well-known [or easily verified] that each H(a/b) is an l-adic Lie group]. Now let us fix a prime number l ∈ Σ † . For S ⊆ X(k) a finite subset, let us denote by (l) Δ X  Δ X (l) Π X  Π X Δ U S  Δ U S ; (l) the maximal pro-l quotients and by Π U S  Π U S ; (l) (l) (l) the quotients of Π U S , Π X by the kernels of Δ U S  Δ U S , Δ X  Δ X . [Here, we recall that Δ U S , Π U S are as defined in Proposition 1.6, (ii), (iii).] Also, for x ∈ X cl , let us write (l) D x (l) [U S ] ⊆ Π U S ; (l) I x (l) [U S ] ⊆ Δ U S for the images of D x [U S ], I x [U S ] [notation as in Proposition 2.1], respectively, (l) in Π U S . Note that we have a natural surjection: (l) (l) (l) Δ U S  Δ X  (Δ X ) ab (l) The cup product on the group cohomology of Δ X determines an isomorphism [cf. Proposition 1.3, (ii)] (l) ∼ (l) (l) Hom((Δ X ) ab , M X ) → (Δ X ) ab (l) def [where we write M X = M X ⊗ Z l ], hence a natural G k -equivariant injection (l) (l) M X → ∧ 2 (Δ X ) ab (l) whose image we denote by I cup . Definition 3.2. We shall refer to the central filtration (l) {Δ U S (m)} (l) (l) (l) on Δ U S with respect to the natural surjection Δ U S  (Δ X ) ab as the weight (l) filtration on Δ U S [cf., e.g., [Mtm], §3, p. 200]. 57 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves Proposition 3.1. (Freeness and Centralizers) Let x ∈ S. Write S x = S\{x}; r for the cardinality of S, g for the genus of X. For x  ∈ S, let (l) ζ x  be a generator of I x  [U S ]. By abuse of notation, we shall also denote by ζ x  (l) the image of ζ x  in Δ U S (2/3). Then: def (l) (i) Gr(Δ U S ) is a free Lie algebra over Z l [hence, in particular, is torsion-free as a Z l -module] which is freely generated by 2g elements (l) α 1 , . . . , α g , β 1 , . . . , β g ∈ Δ U S (1/2) together with the ζ x  ∈ Δ U S (2/3), for x  ∈ S x . Alternatively, for an appropriate (l) (l) choice of the elements ζ x  , Gr(Δ U S ) is the quotient of the free Lie algebra generated by α 1 , . . . , α g , β 1 , . . . , β g , together with the ζ x  ∈ Δ U S (2/3), for x  ∈ S, by the single relation: (l)  ζ x  + x  ∈S g  [α n , β n ] = 0 n=1 At a more intrinsic level, this relation is a generator of the image of the natural G k -equivariant morphism   (l) (l) (l) M X → I x  [U S ] ⊕ I cup x  ∈S (l) ∼ (l) ∼ (l) (l) [determined by the various natural isomorphisms M X → I x  [U S ], M X → I cup ]], (l) whose codomain maps to Gr(Δ U S ) via the natural G k -equivariant mor- phism   (l) (l) (l) I x  [U S ] ⊕ I cup → Δ U S (2/3) x  ∈S (l) (l) [determined by the natural inclusions I x  [U S ] → Δ U S (2/3) and the bracket (l) (l) operation ∧ 2 (Δ X ) ab → Δ U S (2/3)]. (ii) Let ξ be any of the elements α 1 , . . . , α g , β 1 , . . . , β g ; ζ x  , where x  ∈ S x , of (l) (l) (i). Then the centralizer in Gr Q l (Δ U S ) of [the image of] ξ [in Gr Q l (Δ U S )] is (l) equal to Q l · ξ. In particular, the Lie algebra Gr Q l (Δ U S ) is center-free. (l) (iii) Let ξ be as in (ii). Then for m ≥ 1, the centralizer in Δ U S (1/m + 2) of (l) (l) [the image of] ξ [in Δ U S (1/m+2)] is contained in the subgroup of Δ U S (1/m+2) (l) generated by [the image of] ξ and Δ U S (m/m + 2). (iv) Let S ∗ ⊆ S be a subset of S. Write 58 Shinichi Mochizuki (l) (l) New S ∗ ⊆ Gr(Δ U S ) for the sub-Lie algebra over Z l generated by the image of the restriction      (l) (l) (l) I x  [U S ] ⊆ I x  [U S ] → Δ U S (2/3) x  ∈S ∗ x  ∈S to the direct summands indexed by elements of S ∗ of the morphism of (i), and (l) def (l) New S ∗ (a) = Gr(Δ U S )(a) (l) def (l) (l) (l) New S ∗ ; New S ∗ (a/b) = New S ∗ (a)/New S ∗ (b) for (l) a, b ∈ Z such that 1 ≤ a ≤ b. Then, in the notation of (i), New S ∗ is a free Lie algebra over Z l generated by the elements ζ x  , for x  ∈ S ∗ . Moreover, the [“new” and “co-new”] Z l -modules (l) New S ∗ (a/b); def (l) (l) (l) Cnw S ∗ (a/b) = Gr(Δ U S )(a/b)/New S ∗ (a/b) tor,(l) are free. In the following discussion, we shall write New S ∗ Q/Z. def (l) (a/b) = New S ∗ (a/b)⊗ Proof. Assertion (i) (respectively, (ii)) is, in essence, the content of [Kane], Proposition 1 (respectively, Proposition A.1, (ii), (iii)). Assertion (iii) follows formally from assertion (ii). Finally, we consider assertion (iv). By Proposition A.1, (iii), it follows that any free Lie algebra over F l with ≥ 2 generators is center-free. Thus, let M be the module determined by any faithful representa- tion [e.g., when the cardinality of S ∗ is ≥ 2, the adjoint representation] of the free Lie algebra F over F l in the formal generators ζ x  , where x  ∈ S ∗ . Now def (l) observe that we obtain an action of Gr F l (Δ U S ) on M  = M ⊕ M as follows: def We let α 2 , . . . α g ; β 2 , . . . β g ; ζ x  , where x  ∈ S 0 = S\S ∗ , act by multiplication by 0 on M  . We let α 1 , β 1 act on M  = M ⊕ M via the matrices      0 ζ x  0 0 x  ∈S ∗ ; −1 0 0 0 respectively. Finally, we let ζ x  , where x  ∈ S ∗ , act on M  via the following matrix:   0 ζ x  0 −ζ x  Thus, [by assertion (i)] M  determines a representation of Gr F l (Δ U S ) whose (l) (l) (l) restriction to the image of New S ∗ ⊗ Z l F l in Gr F l (Δ U S ) determines [via the (l) natural surjection F  New S ∗ ⊗ Z l F l ] a faithful representation of F . Thus, (l) we conclude that the natural surjection F  New S ∗ ⊗ Z l F l is an isomorphism, (l) (l) and that New S ∗ ⊗ Z l F l injects into Gr F l (Δ U S ). Assertion (iv) now follows formally. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 59 Remark 32. The author wishes to thank A. Tamagawa for pointing out to him the content of Proposition 3.1, (i). Remark 33. One way to verify the existence of the homomorphism “G → Lin(G)(Q l )” of Definition 3.1, (ii), is to think of G as a quotient of a free pro-l group of finite even rank F , whose associated “Gr Q l (−)” is a center- free free Lie algebra [cf. Proposition 3.1, (i), (ii), in the case of r = 1], hence determines an [infinite-dimensional, over Q l ] faithful [cf. Proposition 3.1, (iii)] unipotent representation [i.e., the adjoint representation — cf. the proof of Proposition 3.1, (iv)] of F . More precisely, by Proposition 3.1, (iii), it follows that there exists a unipotent linear representation ρ F : F → GL(V ) on a finite-dimensional Q l -vector space V such that Ker(ρ F ) ⊆ Ker(F  G). But this implies that F  G factors through a quotient F  Q  G such that Q is nilpotent and admits an injective homomorphism of topological groups ρ Q : Q → Q alg (Q l ) [induced by ρ F ], where Q alg is a connected, unipotent algebraic group over Q l , such that ρ Q is a local isomorphism, and Ker(ρ Q ) ⊆ Ker(Q  G). Thus, ρ Q determines a structure of l-adic Lie group on Q such that the morphism Lie(ρ Q ) induced by ρ Q on Lie algebras is an isomorphism. Moreover, the morphism induced by Q  G on Lie algebras factors through Lie(ρ Q ), thus determining a homomorphism of [connected, unipotent] algebraic groups Q alg → Lin(G) such that the resulting composite homomorphism Q → Q alg (Q l ) → Lin(G)(Q l ) factors [cf. the induced morphisms on Lie algebras, together with the fact that Lin(G)(Q l ) has no torsion!] though G, thus yielding a homomorphism G → Lin(G)(Q l ), as desired. Next, let us fix an x ∗ ∈ S, as well as a choice of decomposition group D x ∗ [U S ] ⊆ Π U S [i.e., among the various Π U S -conjugates of this subgroup] associated to x ∗ . [Thus, D x ∗ [U S ] determines a specific subgroup [i.e., not just a conjugacy class (l) (l) of subgroups] D x ∗ [U S ] ⊆ Π U S .] Recall that the natural exact sequences 1 → I x ∗ [U S ] → D x ∗ [U S ] → G k → 1; 1 → I x (l) [U S ] → D x (l) ∗ [U S ] → G k → 1 ∗ split. [Indeed, extracting roots of any local uniformizer of X at x ∗ determines such a splitting — cf., e.g., the discussion at the beginning of [Mzk8], §4.] In the following discussion, we shall fix a splitting G k → D x ∗ [U S ] of this exact sequence. Thus, this splitting determines a natural action of G k (l) [by conjugation] on Δ U S , hence also on (l) def (l) (l) def (l) Lin U S (a/b) = Lin(Δ U S (a/b))(Q l ); Lie U S (a/b) = Lie(Δ U S (a/b)) (l) Gr Q l (Δ U S )(a/b) 60 Shinichi Mochizuki [where a, b ∈ Z; 1 ≤ a ≤ b]. Write F k ∈ G k for the Frobenius element of G k . In the following, we shall denote the cardinality of k by q k . (Galois Invariant Splitting) Let a, b ∈ Z, 1 ≤ Proposition 3.2. a ≤ b. (l) (i) The eigenvalues of the action of F k on Lie U S (a/a+1) are algebraic numbers a/2 all of whose complex absolute values are equal to q k [i.e., “of weight a”]. (ii) There is a unique G k -equivariant isomorphism of Lie algebras ∼ (l) (l) Lie U S (a/b) → Gr Q l (Δ U S )(a/b) ∼ (l) (l) which induces the identity isomorphism Lie U S (c/c + 1) → Gr Q l (Δ U S )(c/c + 1), for all c ∈ Z such that a ≤ c ≤ b − 1. (l) (iii) The isomorphism of (ii) together with the natural inclusions I x [U S ] → (l) (l) Δ U S for x ∈ S [which are well-defined up to Δ U S -conjugation] determine a G k -equivariant morphism   (l) (l) I x (l) [U S ] ⊗ Q l ⊕ Lie U S (1/2) → Lie U S (1/∞) x∈S (l) which exhibits, in a G k -equivariant fashion, Lie U S (1/∞) as the quotient of the completion [with respect to the filtration topology] of the free Lie algebra generated by the finite dimensional Q l -vector space   (l) I x (l) [U S ] ⊗ Q l ⊕ Lie U S (1/2) x∈S [equipped with a natural grading, hence also a filtration, by taking the (l) (l) I x [U S ] ⊗ Q l to be of weight 2, Lie U S (1/2) to be of weight 1], by the single relation determined by the image of the morphism   (l) (l) I x (l) [U S ] ⊗ Q l ⊕ (I cup ⊗ Q l ) M X ⊗ Q l → x∈S of Proposition 3.1, (i), tensored with Q l . (l) (l) (iv) For each g ∈ Lin U S (1/∞), there exists a unique h ∈ Lin U S (1/∞) such Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 61 that F k ◦ Inn g = Inn h ◦ F k ◦ Inn h −1 (l) [where “Inn” denotes the inner automorphism of Lin U S (1/∞) defined by con- jugation by the subscripted element]. Moreover, when g lies in the image of (l) I x ∗ ⊗ Q l [which is stabilized by the action of F k ], h also lies in the image of (l) I x ∗ ⊗ Q l . Proof. Assertion (i) follows immediately from the “Riemann hypothesis for abelian varieties over finite fields” — cf., e.g., [Mumf], p. 206. Assertion (ii) (respectively, (iii); (iv)) follows formally from assertion (i) (respectively, and Proposition 3.1, (i); and successive approximation of h with respect to the (l) (l) natural filtration Lin U S (a/∞) ⊆ Lin U S (1/∞)). Next, let S ∗ ⊆ S def be a subset such that x ∗ ∈ S ∗ ; S 0 = S\S ∗ . In the following, we shall regard (l) Lin U S (a/b) as being equipped with its natural l-adic topology. Thus, G k acts (l) (l) continuously on Lin U S (a/b), Lie U S (a/b), and we have natural G k -equivariant surjections: (l) (l) Lin U S (a/b)  Lin U S (a/b); 0 (l) (l) Lie U S (a/b)  Lie U S (a/b) 0 Let us write (l) Lin U S /U S (a/b); 0 (l) Lie U S /U S (a/b) 0 for the kernels of these surjections. In the following, to simplify the notation, we shall often omit the superscript (l) from the objects “Lin (l) ”, “Lie (l) ”, “New (l) ”, “New tor,(l) ” introduced above and write: Lin U S (a/b); Lin U S /U S 0 (a/b); Lie U S (a/b); Lin U S 0 (a/b); Lie U S 0 (a/b) Lie U S /U S 0 (a/b); New S ∗ (a/b); New tor S ∗ (a/b) Also, we shall write: New Q S ∗ (a/b) = New S ∗ (a/b) ⊗ Q; def Note that, for Z def Δ Lie U S = Lin U S (1/∞) × Lin US (1/∞) Δ U S 0 0 b ≥ 1, we have a natural G k -equivariant inclusion ∼ Lin U S /U S 0 (b + 1/∞) → Lin U S /U S 0 (b + 1/∞) × {1} {1} → Lin U S (1/∞) × Lin US (1/∞) Δ U S 0 = Δ Lie U S 0 62 Shinichi Mochizuki whose image forms a normal subgroup of Δ Lie U S ; write Lie≤b Δ Lie U S  Δ U S for the quotient of Δ Lie U S by this normal subgroup. Also, we have a natural G k -equivariant [composite] inclusion ∼ Lie≤b+1 New Q S ∗ (b+1/b+2) → Lie U S /U S 0 (b+1/b+2) → Lin U S /U S 0 (b+1/b+2) → Δ U S Lie≤b+1 whose image forms a normal subgroup of Δ U ; write S Lie≤b+1 Lie≤b+  Δ U Δ U S S for the quotient of Δ Lie≤b+1 by this normal subgroup. Thus, we have natural U S G k -equivariant homomorphisms of topological groups: Lie≤b+ Lie≤b Δ U S → Δ Lie  Δ U  Δ U S 0 U S  Δ U S S [the last three of which are easily verified to be surjective]. Moreover, forming the semi-direct product with G k [via the natural actions of G k ] yields topological groups and homomorphisms as follows: Lie≤b+ Lie≤b  Π U  Π U S 0 Π U S → Π Lie U S  Π U S S Also, we note that we have natural exact sequences: 1 → Lin U S /U S 0 (1/∞) → Δ Lie U S → Δ U S 0 → 1 1 → Lin U S /U S 0 (1/∞) → Π Lie U S → Π U S 0 → 1 Definition 3.3. Lie≤b Lie≤b Lie≤b+ Lie (i) We shall refer to Δ Lie ; Π U ; Δ U ; Π Lie≤b+ ) U S (respectively, Π U S ; Δ U S U S S S as the [l-adic] Lie-ification (respectively, Lie-ification; Lie-ification, truncated to order b; Lie-ification, truncated to order b; Lie-ification, truncated to order b+; Lie-ification, truncated to order b+) of Δ U S (respectively, Π U S ; Δ U S ; Π U S ; Δ U S ; Π U S ) [over Δ U S 0 (respectively, Π U S 0 ; Δ U S 0 ; Π U S 0 ; Δ U S 0 ; Π U S 0 )]. (ii) Observe that it follows immediately from the definitions that, for Z we have natural exact sequences Lie≤b+1 1 → New Q → Δ Lie≤b+ → 1 S ∗ (b + 1/b + 2) → Δ U S U S Lie≤b+1 → Π Lie≤b+ → 1 1 → New Q U S S ∗ (b + 1/b + 2) → Π U S b ≥ 1, Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 63 on which Π Lie≤b+1 acts naturally by conjugation. [Here, we note in passing U S that it is immediate from the definitions that the submodule New S ∗ (b + 1/b + 2) ⊆ New Q S ∗ (b + 1/b + 2) is contained in the image of Δ U S .] In particular, we obtain a natural inclusion: Lie≤b+1 Lie≤b+1 New S ∗ (b + 1/b + 2) → Δ U (⊆ Π U ) S S Lie≤b+1 We shall refer to the quotients of Δ Lie≤b+1 , Π U by the image of this U S S tor≤b+1 tor≤b+1 natural inclusion as the toral Lie-ifications Δ U S , Π U of Δ U S , Π U S S [over Δ U S 0 , Π U S 0 ]. Thus, we have natural exact sequences tor≤b+1 Lie≤b+ 1 → New tor → Δ U → 1 S ∗ (b + 1/b + 2) → Δ U S S tor≤b+1 → Π Lie≤b+ → 1 1 → New tor S ∗ (b + 1/b + 2) → Π U S U S Lie≤b+1 acts naturally by conjugation. on which Π U S (iii) Suppose that U S   → U S 0 is a connected finite étale covering that arises 0 from an open subgroup Π U   ⊆ Π U S 0 ; write X  → X for the normalization of S 0 X in U S   . Then we shall say that the [ramified] covering X  → X is (S, S 0 , Σ)- 0 admissible if every closed point of X  that lies over a point of S is rational over the base field k  of X  , and, moreover, Π U   is a characteristic subgroup of S Π U S 0 . 0 Remark 34. Note that it follows immediately from the definition of [cf. also Proposition 3.2, (iii)] that we obtain a natural subgroup Π Lie U S def = D x Lie ∗   I x (l) [U ] ⊗ Q  G k ⊆ Π Lie S U S ∗ which contains the image of the decomposition group D x ∗ [U S ] ⊆ Π U S via b ≥ 1, the natural homomorphism Π U S → Π Lie U S . Let us write, for Z def def Lie≤b Lie≤b Lie≤b D x Lie≤b Δ Lie ⊆ Π U for the image of D x Lie in Π U ; I x Lie = D x Lie = U S ; I x ∗ ∗ ∗ ∗ ∗ S S Lie≤b Lie≤b D x ∗ Δ U S . [Also, we shall use similar notation when “b” is replaced by “b+”.] Proposition 3.3. free. (Center-freeness of Lie-ification) Δ Lie U S is center- 64 Shinichi Mochizuki Proof. Since Δ U S 0 is center-free [cf. Proposition 1.6, (iii)], and the nat- ural morphism Δ Lie U S → Δ U S 0 is surjective, it suffices to verify that the cen- Lie tralizer in Lin U S (1/∞) of the image of Δ Lie U S is trivial. But the image of Δ U S in Lin U S (1/∞) contains the image of Δ U S in Lin U S (1/∞). In particular, it follows that the centralizer in question lies in the center of Lin U S (1/∞). Thus, Proposition 3.3 follows from Proposition 3.1, (ii) [or, alternatively, (iii)]. Remark 35. Observe that changing the choice of splitting G k → D x ∗ [U S ] affects the image of the element F k ∈ G k via the composite of the inclusion G k → Π U S with the morphisms Π U S → Π Lie U S ; Lie≤b Π U S → Π U ; S Π U S → Π Lie≤b+ U S by conjugation by an element h ∈ I x Lie , which, up to a denominator divid- ∗ ing q k − 1, lies in the image of I x ∗ [U S ] ⊆ Δ U S — cf. Proposition 3.2, (iv); Proposition 3.3. In particular, it follows that changing the choice of split- tings G k → D x ∗ [U S ] affects the Galois invariant splittings of Proposition 3.2, (ii), by conjugation by h. Put another way, if we identify the “Lin U S (1/∞)”, Lie “Lin U S 0 (1/∞)” portions of Δ Lie U S [cf. the definition of Δ U S ] with the [l-adic points of the pro-unipotent algebraic groups determined by the] correspond- ing graded Lie objects “Gr Q l (−)(1/∞)” via the Galois invariant splittings of Proposition 3.2, (ii), then it follows that: Changing the choice of splitting G k → D x ∗ [U S ] affects the images of the morphisms Π U S → Π Lie U S ; [where Z Lie≤b Π U S → Π U ; S Π U S → Π Lie≤b+ U S b ≥ 1] by conjugation by h. In light of Proposition 3.3, we may apply the exact sequence “1 → (−) → Aut(−) → Out(−) → 1” [cf. §0] to construct the following topological group: Aut(Δ Lie Gal(X k  /X k ) Δ LIE U S = lim U   ) × Out(Δ Lie  ) ←− def X  S U  S [where X  → X ranges over the (S, S 0 , Σ)-admissible coverings of X; U S   ⊆ X  is the open subscheme determined by the complement of the set S  of closed points of X  that lie over points of S]. Note that G k acts naturally on Δ LIE U S ; with G to obtain a topo- thus, we may form the semi-direct product of Δ LIE k U S   logical group Π LIE U S . Also, since the various Δ U   [where U S  ⊆ X is the open S 0 0 subscheme determined by the complement of the set S 0  of closed points of X  65 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves that lie over points of S 0 ] arising from the X  → X that appear in this inverse limit are center-free [cf. Proposition 1.6, (iii)], the natural isomorphism ∼ Aut(Δ U   ) × Out(Δ U  ) Gal(X k  /X k ) → Δ U S 0 lim ←− X   S 0 S 0 LIE determines surjections Δ LIE U S  Δ U S 0 , Π U S  Π U S 0 . Next, let us observe that, for Z b ≥ 1, the various quotients Δ Lie U    S tor≤b+1 Lie≤b Δ U  Δ Lie≤b+  Δ U determine quotients of topological groups Δ LIE   U S  U  S  S  S  TOR≤b+1 Δ TOR≤b+1  Δ LIE≤b+  Δ LIE≤b , Π LIE  Π LIE≤b+  Π LIE≤b . U S  Π U S U S U S U S U S U S Thus, we obtain natural homomorphisms of topological groups: TOR≤b+1 Δ U S → Δ LIE  Δ LIE≤b+  Δ LIE≤b  Δ U S 0 U S  Δ U S U S U S TOR≤b+1 Π U S → Π LIE  Π LIE≤b+  Π LIE≤b  Π U S 0 U S  Π U S U S U S We shall denote by ≤b+ ⊆ Δ LIE≤b+ ; Δ U U S S ≤b+ Π U ⊆ Π LIE≤b+ ; U S S LIE≤b Δ ≤b ; U S ⊆ Δ U S LIE≤b Π ≤b U S ⊆ Π U S the respective images of Δ U S , Π U S via these natural homomorphisms. Thus, ≤b one may think of Δ ≤b U S , Π U S as being a sort of “canonical integral structure” on the “inverse limit truncated Lie-ifications” Δ LIE≤b , Π LIE≤b . U S U S Here, we note in passing, relative to the theory of §1, 2, that [it is immediate from the definitions that] when S = S ∗ [so U S 0 = X], the quotient Π U S  Π ≤2 U S is the maximal cuspidally pro-l abelian quotient of Π U S [cf. Proposition 1.9, (i)]. LIE Next, let us observe that in the inverse limit used to define Δ LIE U S , Π U S , Lie Lie the various “I x ∗ ”, “D x ∗ ” [cf. Remark 34] form a compatible system, hence give rise to subgroups ⊆ D x LIE ⊆ Π LIE I x LIE U S ; ∗ ∗ I x LIE≤b ⊆ D x LIE≤b ⊆ Π LIE≤b U S ∗ ∗ together with natural exact sequences and isomorphisms [when b ≥ 2] → D x LIE → G k → 1 1 → I x LIE ∗ ∗ 1 → I x LIE≤b → D x LIE≤b → G k → 1 ∗ ∗ ∼ ∼ I x LIE = I x LIE≤b = I x ∗ [U S ] ⊗ Q ∗ ∗ (l) [and similarly when “b” is replaced by “b+”]. Also, the images of the subgroups I x ∗ [U S ], D x ∗ [U S ] of Π U S determine subgroups ⊆ D x ≤b ⊆ Π ≤b I x ≤b U S ∗ ∗ [and similarly when “b” is replaced by “b+”]. 66 Shinichi Mochizuki In the following, let us write [cf. Proposition 3.1, (iv)] def (l) Cnw S ∗ (a/b) = Cnw S ∗ (a/b); Cnw Q S ∗ (a/b) = Cnw S ∗ (a/b) ⊗ Q def (l) [where a, b ∈ Z, 1 ≤ a ≤ b]. Before proceeding, let us observe that [it is immediate from the definitions that] the natural surjections Δ LIE≤1+  Δ LIE≤1  Δ U S 0 ; U S U S Π LIE≤1+  Π LIE≤1  Π U S 0 U S U S are isomorphisms. On the other hand, for b ≥ 2, we have the following result: Proposition 3.4. tures) For Z b ≥ 2: (Plus Liftings of Canonical Integral Struc- ≤b+ ≤b+ ≤b  Δ ≤b (i) The natural surjections Δ U U S , Π U S  Π U S are isomorphisms. S LIE≤b to inclusions (ii) Any two liftings of the natural inclusion Π ≤b U S → Π U S LIE≤b+ differ by conjugation in Π LIE≤b+ by a unique element of Π ≤b U S → Π U S U S LIE≤b+ the kernel of Π U S  Π LIE≤b . U S LIE≤b to inclusions (iii) Any two liftings of the natural inclusion Π ≤b U S → Π U S LIE≤b+ whose images contain D x ≤b+ in fact coincide. Π ≤b U S → Π U S ∗ Proof. First, we consider assertion (i). It follows immediately from the definitions that the kernel in question ≤b+ ≤b+ ≤b Ker(Δ U  Δ ≤b U S ) = Ker(Π U S  Π U S ) S is contained in [and, in fact, equal to] the inverse limit lim Cnw S ∗  (b + 1/b + 2) ←− X  [where X  → X ranges over the (S, S 0 , Σ)-admissible coverings of X; S ∗  (respec- tively, S  ) is the set of closed points of X  that lie over points of S ∗ (respectively, S)]. On the other hand, it follows from the definition of “Cnw S ∗  (b + 1/b + 2)” that Cnw S ∗  (b + 1/b + 2) is generated by certain successive brackets of the var- (l) ious generators of the Lie algebra Gr(Δ U  ) [cf. Proposition 3.1, (i)] with the S  property that at least one of the generators appearing in the successive bracket is [in the notation of Proposition 3.1, (i)] either one of the [analogue for X  def of the] “α 1 , . . . , α g , β 1 , . . . , β g ” or one of the “ζ x  ”, where x  ∈ S 0  = S  \S ∗  . Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 67 Moreover, since, by taking Π U   ⊆ Π U   to be sufficiently small, one may ar- S 0 (l) S 0 (l) range that the image of Δ U  (1/3) in Δ U  (1/3) be contained in an arbitrarily  S 0  S 0 (l) small open subgroup of Δ U  (1/3), it thus follows that the above inverse limit S  0 vanishes. This completes the proof of assertion (i). Next, let us observe that to prove assertion (ii), it suffices — in light of the natural isomorphism ∼  Π LIE≤b ) → lim Cnw Q Ker(Π LIE≤b+ U S U S S ∗  (b + 1/b + 2) ←− X  [where X  , S ∗  are as above] — to show that Q H i (Π ≤b U S , Cnw S ∗  (b + 1/b + 2)) = 0 Q for i = 0, 1, each S ∗  as above. Since the action of Δ ≤b U S on Cnw S ∗  (b + 1/b + 2) clearly factors through a finite quotient of Δ ≤b U S  Δ U S 0 , it thus suffices to observe [by considering the Leray spectral sequence associated to the surjection Q Π ≤b U S  G k ] that the action of F k on Cnw S ∗  (b+ 1/b + 2) is “of weight b + 1 ≥ 3”, (l) while the action of F k on (Δ U  ) ab is “of weight ≤ 2” [cf. Proposition 3.2, (i)]. S  This completes the proof of assertion (ii). Finally, we consider assertion (iii). First, let us observe that any two liftings LIE≤b LIE≤b+ to inclusions Π ≤b whose of the natural inclusion Π ≤b U S → Π U S U S → Π U S ∼ ≤b+ ≤b ≤b images contain D x ∗ → D x ∗ [since b ≥ 2] in fact coincide on D x ∗ ⊆ Π ≤b U S . Thus, by assertion (ii), it suffices to verify that the submodule of F k -invariants of  Π LIE≤b ) Ker(Π LIE≤b+ U S U S is zero. But in light of the natural isomorphism ∼ Ker(Π LIE≤b+  Π LIE≤b ) → lim Cnw Q S ∗  (b + 1/b + 2) U S U S ←− X  [where X  , S ∗  are as above], this follows from Proposition 3.2, (i). This com- pletes the proof of assertion (iii). Next, for Z b ≥ 1, let us denote by ⊆ Δ TOR≤b+1 ; Δ ≤b++ U S U S ≤b++ TOR≤b+1 Π U ⊆ Π U S S the respective images of Δ U S , Π U S via the natural homomorphisms considered above and by ≤b++ I x ≤b++ ⊆ D x ≤b++ ⊆ Π U ∗ ∗ S 68 Shinichi Mochizuki the images of the subgroups I x ∗ [U S ], D x ∗ [U S ] of Π U S . Observe that it follows TOR≤b+1 , Π TOR≤b+1 [cf. also Proposition 3.1, (iv)] from the definition of Δ U U S S ≤b++ ≤b+ ≤b++ that the natural surjections Δ U S  Δ U S , Π U  Π ≤b+ are, in fact, U S S isomorphisms. Thus, by Proposition 3.4, (i), we obtain a commutative diagram of natural homomorphisms ≤b+1 Π U S ⏐ ⏐ Π LIE≤b+1 U S  ≤b++ Π U S ∼ → ≤b+ Π U S ∼ → ⏐ ⏐ ⏐ ⏐ ⏐ ⏐  Π TOR≤b+1 U S  Π LIE≤b+ U S Π ≤b U S  Π LIE≤b U S [where the vertical arrows are the natural inclusions; all of the horizontal arrows are surjections; the second two upper horizontal arrows are isomorphisms]. Moreover, it follows immediately from the definitions that the first square in ⊆ this commutative diagram is cartesian. That is to say, the subgroup Π ≤b+1 U S LIE≤b+1 Π U may be thought of as the inverse image via the natural surjection S LIE≤b+1  Π TOR≤b+1 of the image of a certain lifting of the natural inclusion Π U U S S ≤b LIE≤b+ TOR≤b+1 Π U S → Π U S [cf. Proposition 3.4, (i)] to an inclusion Π ≤b . U S → Π U S Also, let us write: ≤b Π ≤b U S [csp] = Ker(Π U S  Π X ) def ≤b++ ≤b++ [csp] = Ker(Π U  Π X ) Π U S S def ≤b++ for the cuspidal subgroups of Π ≤b . U S , Π U S Next, following the pattern of §1, we relate the constructions made so far to the fundamental groups Δ U X×X , Π U X×X [cf. the discussion preceding Proposition 1.5]. For simplicity, we assume from now on that: S = S ∗ = {x ∗ } [so S 0 = ∅]. Write D x ∗ [X] ⊆ Π X for the image of D x ∗ [U S ] via the natural surjection Π U S  Π X . Then the projection Π U X×X  Π X to the second factor determines a natural isomorphism ∼ Π U S → Π U X×X × Π X D x ∗ [X] [cf. Proposition 1.6, (ii)]. Moreover, this isomorphism determines a natural isomorphism (Π U S ⊇) ∼ D x ∗ [U S ] → D X × Π X D x ∗ [X] (⊆ D X ⊆ Π U X×X ) [where “D X ” is as in the discussion preceding Proposition 1.8] which is compat- ible with the natural inclusions D x ∗ [U S ] → Π U S , D X → Π U X×X . Put another way, D x ∗ [U S ] [hence also I x ∗ [U S ], G k ⊆ D x ∗ [U S ]] may be thought of as being Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 69 “simultaneously” a subgroup of both Π U S and D X . Thus, we obtain a natural exact sequence 1 −→ Δ U S −→ Π U X×X −→ Π X −→ 1 together with compatible inclusions Δ U S ⊇ Δ U S   ⊇ I x ∗ [U S ] ⊆ D x ∗ [U S ] ⊆ D X ⊆ Π U X×X [where X  → X is an (S, ∅, Σ)-admissible covering of X; U S   ⊆ X  is the open subscheme determined by the complement of the set S  of closed points of X  that lie over x ∗ ]. Also, we shall write:  Δ def Δ U X×X ⊆ Π U X×X = D X D X In particular, we obtain natural actions [by conjugation] of D X on Δ U S , Δ U S   [as well as on the various objects naturally constructed from Δ U S , Δ U S   in the above discussion], which we shall refer to as diagonal actions. Proposition 3.5. (Characterization of the Diagonal Action) Suppose that S = S ∗ = {x ∗ }. Then in the notation and terminology of the above discussion, the diagonal action of D X on Lin U S   (1/∞) is completely determined [i.e., as a continuous action of the topological group D X on the topological group Lin U S   (1/∞)] by the following conditions: (a) the action is compatible with the natural action of D x ∗ [U S ] ⊆ D X on Lin U S   (1/∞); (b) the action is compatible with the filtration {Lin U S   (a/∞)} [where a ≥ 1 is an integer] on Lin U S   (1/∞). (c) the action coincides with the diagonal action of D X on the quotient Lin U S   (1/4) [cf. condition (b)] of Lin U S   (1/∞). Proof. First, I claim that it suffices to show that these conditions deter- def Δ Δ Δ mine the action of the subgroup D X  /X = D X × Δ X Δ X  ⊆ D X ⊆ D X on Δ Lin U S   (1/∞). Indeed, once the action of D X  /X is determined, it follows that the action of def Δ D X  /X = D X × Π X Π X  ⊆ D x ∗ [U S ] · D X  /X ⊆ D X is determined [cf. condition (a)]. On the other hand, since Π X  is an open normal subgroup of Π X , it follows that D X  /X is an open normal subgroup of 70 Shinichi Mochizuki D X . Thus, by considering the conjugation actions of D X on D X  /X and of Im(D X ) ⊆ Lin U S   (1/∞) on Im(D X  /X ) ⊆ Lin U S   (1/∞) [i.e., of the group of automorphisms of Lin U S   (1/∞) induced by elements of D X on the group of automorphisms of Lin U S   (1/∞) induced by elements of D X  /X ], we conclude that the action of D X on Lin U S   (1/∞) is determined up to composition with automorphisms of Lin U S   (1/∞) that commute with the action of D X  /X and [cf. condition (c)] induce the identity on the quotient Lin U S   (1/4). Now let α be an automorphism of Lin U S   (1/∞) that commutes with the action of D X  /X and induces the identity on the quotient Lin U   (1/4). Then α commutes with S some open subgroup of G k ⊆ D x ∗ [U S ] ⊆ D X , so α induces an automorphism of Lie U S   (1/∞) that is compatible with the splittings of Proposition 3.2, (ii). Since (l) Gr(Δ U  ) is generated by its elements “of weight ≤ 2” [cf. Proposition 3.1, (i)], S  we thus conclude that α induces the identity automorphism of Lie U S   (1/∞), hence that α itself is the identity automorphism. This completes the proof of the claim. Next, let us observe that by condition (c) [cf. also Proposition 3.1, (i)], Δ the action of D X  /X on Lin U   (1/∞) is unipotent, relative to the filtration of S condition (b). Thus, it follows [from the definition of “Lie(−)”] that the induced Δ action of D X  /X on Lie U   (1/∞) determines an action of the Lie algebra S def Δ Δ (l) Lie(D X (1/∞))  /X ) = Lie((D X  /X ) Δ (l) Δ (l) for the maximal pro-l quotient of (D X ] on [where we write (D X  /X )  /X ) the Lie algebra Lie U S   (1/∞). Moreover, to complete the proof of Proposition 3.5, it suffices to show that this Lie algebra action is the action arising from the diagonal action. In fact, since this Lie algebra action is compatible [cf. Δ condition (a)] with the actions of G k on Lie(D X  /X ), Lie U   (1/∞), it follows, S by considering the induced eigenspace splittings [cf. Proposition 3.2, (ii)], that [to complete the proof of Proposition 3.5] it suffices to show that the Lie algebra def (l) Δ Δ action of Gr(D X  /X ) = Gr(Lie(D X  /X )) on Gr(Δ U  ) is the action arising S  (l) Δ from the diagonal action. On the other hand, since Gr(D X  /X ), Gr(Δ U  ) are S  generated by elements “of weight ≤ 2” [cf. Proposition 3.1, (i)], this follows Δ by observing that the Lie algebra action of the unique generator of Gr(D X  /X ) Δ “of weight 2” [which arises from I x ∗ [U S ] ⊆ D X  /X ] is determined by condition Δ (a), while the Lie algebra action of the generators of Gr(D X  /X ) “of weight (l) (l) S  S  1” [which send elements of Gr(Δ U  ) “of weight ≤ 2” to elements of Gr(Δ U  ) “of weight ≤ 3”] is determined by condition (c). This completes the proof of Proposition 3.5. Remark 36. Note that the conditions of Proposition 3.5 allow one to characterize not only the diagonal action of D X on Lin U S   (1/∞), but also Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 71 Lie LIE on Δ Lie , Π LIE  , Π U  , hence also on Δ U U S [where we note that the diagonal U S S  S   action of D X on Gal(X k /X k ) is simply the conjugation action arising from the quotients D X  Π X , Δ X  Gal(X k  /X k )]. Remark 37. Note that the groups Lin U S   (1/4) of condition (c) of Proposition 3.5 are, as groups equipped with the surjection Lin U S   (1/4)  Lin X  (1/4), cuspidally abelian [i.e., the kernel of this surjection is abelian], hence may be constructed from the maximal cuspidally abelian quotients Π U X×X  Π c-ab U X×X of Theorem 1.1. Proposition 3.6. (Extensions of Canonical Integral Structures) Suppose that S = S ∗ = {x ∗ } [cf. Remark 39 below]. Let b ≥ 1 be an integer. Then: (i) Suppose that b = 1. Then any two liftings of the natural inclusion Π ≤b U S → LIE≤b+ TOR≤b+1 Π U to inclusions Π ≤b differ by conjugation in Π TOR≤b+1 U S → Π U S U S S LIE≤b+ by an element of the kernel of Π TOR≤b+1  Π . U S U S (ii) Suppose that b ≥ 2. Then any two liftings of the natural inclusion Π ≤b U S → LIE≤b+ TOR≤b+1 Π U to inclusions Π ≤b whose images contain I x ≤b++ differ U S → Π U S ∗ S TOR≤b+1 TOR≤b+1 by an element of the kernel of Π U  by conjugation in Π U S S LIE≤b+ . Π U S ≤b+1 that satisfies the (iii) Let β be an automorphism of the profinite group Π U S following two conditions: (a) β preserves and induces the identity on the quo- ≤b+1 ≤b+1  Π ≤b ⊆ Π ≤b+1 . Then β is tient Π U U S ; (b) β preserves the subgroup I x ∗ U S S ≤b+1 ≤b a Ker(Π U  Π )-inner automorphism. U S S TOR≤b+1 (iv) Let δ ∈ Ker(Π U  Π LIE≤b+ ) be an element that is invariant un- U S S der the diagonal action of D X . Then if b = 1, then δ lies in the image of I x ∗ [U S ] ⊗ (Q l /Z l ); if b ≥ 2, then δ is the identity element. (v) Write Π ≤b Π U S  Π ≤∞ U S = lim U S ; ←− def Δ U S  Δ ≤∞ Δ ≤b U S = lim U S ←− def b b ≤b for the the quotients of Π U S , Δ U S defined by the inverse limit of the Π ≤b U S , Δ U S and Π U X×X  Π ≤∞ U X×X ; Δ U X×X  Δ ≤∞ U X×X 72 Shinichi Mochizuki for the quotients of Π U X×X , Δ U X×X determined by the kernel in Δ U S ⊆ Δ U X×X ⊆ Π U X×X [cf. the discussion preceding Proposition 3.5] of Ker(Δ U S  Δ ≤∞ U S ). ≤∞ ≤∞ Then Π U S  Π ≤∞ (respectively, Δ  Δ ; Π  Π ; Δ U S U X×X U X×X  U S U S U X×X Δ ≤∞ U X×X ) is the maximal cuspidally pro-l quotient of Π U S (respectively, ≤∞ ≤∞ ≤∞ † † Δ U S ; Π U X×X ; Δ U X×X ); moreover, (Π ≤∞ U S ) , Δ U S , (Π U X×X ) , Δ U X×X [where the daggers denote the result of applying the operation “× G k G † k ”] are slim. Proof. First, we consider assertions (i), (ii). Observe that, for Z b ≥ 1, LIE≤b+ the difference of any two liftings of the natural inclusion Π ≤b to U S → Π U S TOR≤b+1 inclusions Π ≤b → Π determines a compatible collection of cohomology U S U S classes tor η S  ∈ H 1 (Π ≤b U S , New S  (b + 1/b + 2)) [where X  → X ranges over the (S, ∅, Σ)-admissible coverings of X; S  is the set of closed points of X  that lie over x ∗ ]. Since New tor S  (b + 1/b + 2) = 0 whenever b is even, we may assume for the remainder of the proof of assertions (i), (ii) that b is odd. Next, let us observe that by Proposition 3.2, (i), the zeroth cohomology module tor H 0 (Π ≤b U S , New S  (b + 1/b + 2)) is finite. This finiteness implies that any [not necessarily compatible!] system tor of sections of a compatible system of torsors over H 0 (Π ≤b U S , New S  (b + 1/b + 2)) always admits a compatible cofinal subsystem. In light of the natural isomor- phism ∼ TOR≤b+1  Π LIE≤b+ ) → lim New tor Ker(Π U S  (b + 1/b + 2)) U S S ←− X  [where X  , S  are as described above], we thus conclude that in order to show TOR≤b+1 that the two inclusions Π ≤b differ by conjugation by an element U S → Π U S TOR≤b+1 LIE≤b+  Π U S ), it suffices to show that the η S  = 0. of Ker(Π U S Note that Π ≤b [csp] acts trivially on New tor S  (b+ 1/b + 2)). Now I claim that: U S Each η S  arises from a unique class [which, by abuse of notation, we shall also denote by η S  ] in H 1 (Π X , New tor S  (b + 1/b + 2)). Indeed, if b = 1, this claim follows from the fact that Π ≤b U S [csp] = {1} [cf. the discussion preceding Proposition 3.4], so assume that b ≥ 2, and that we are in the situation of assertion (ii). Now observe that since S = S ∗ is of cardinality ≤b++ [csp]) is topologically generated one, it follows that Π ≤b U S [csp] (respectively, Π U S ≤b++ - (respectively, Π -) conjugates of I x ≤b (respectively, I x ≤b++ ). by the Π ≤b U S U S ∗ ∗ Note, moreover, that it is immediate from the definitions that every element  Π LIE≤b+ ) commutes with I x ≤b++ . In particular, it follows of Ker(Π TOR≤b+1 U S U S ∗ 73 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves ≤b TOR≤b+1 that the images of Π ≤b under U S [csp] via the two inclusions Π U S → Π U S consideration necessarily coincide. But this implies that each η S  arises from a unique class in H 1 (Π X , New tor S  (b + 1/b + 2)), thus completing the proof of the claim. Next, [returning to the general situation involving both assertions (i) and (ii)] let X  → X  be a morphism of (S, ∅, Σ)-admissible coverings of X. Write U S   ⊆ X  for the open subscheme determined by the complement of the set S  of closed points of X  that lie over points of S. Also, let us assume that the open subgroup Δ X  ⊆ Δ X  arises from some open subgroup H  ⊆ Δ ab X  that is preserved by the action of Π X . Thus, it follows that the covering X k  → X k  is abelian; write def Gal(X  /X  ) = Gal(X k  /X k  ). For c a positive integer, set: R  = Z l ; def R c  = Z l [c · Gal(X  /X  )] ⊆ R  = Z l [Gal(X  /X  )] def def [where we write c · Gal(X  /X  ) ⊆ Gal(X  /X  ) for the subgroup of the abelian group Gal(X  /X  ) that arises as the image of multiplication by c]. Thus, R  (respectively, R  ; R c  ) is a commutative ring with unity whose underlying R  - (respectively, R c  -; R  -) module is finite and free; moreover, R  , R c  admit a natural Π X -action [induced by the conjugation action of Π X on the subquotient Gal(X  /X  ) of Π X ]. Also, we shall denote by   c : R c   R  ;   : R   R  the augmentations obtained by mapping all of the elements of Gal(X  /X  ) to 1. Next, let us observe that S  , S  admit natural Π X -actions with respect to which we have natural isomorphisms of Π X -modules [cf. Proposition 3.1, (i), (iv)] ∼ New S  (2/3) → R  [S  ] ⊗ M X ; (l) ∼ New S  (2/3) → R  [S  ] ⊗ M X (l) which determine natural isomorphisms of Π X -modules ∼ (l) ∼ (l) New S  (2c/2c + 1) → Lie cR  (R  [S  ] ⊗ M X ) New S  (2c/2c + 1) → Lie cR  (R  [S  ] ⊗ M X ) [cf. the notation of Proposition A.1] for integers c ≥ 1. In the following, we shall identify the domains and codomains of these isomorphisms via these isomorphisms. Next, let us observe that the R  -module R  [S  ] admits a natural R  -module structure that is compatible with the Π X -action on R  , R  [S  ]. Note, moreover, that R  [S  ] is a free R  -module, and that we have a natural isomorphism ∼ R  [S  ] ⊗ R  ,  R  → R  [S  ] 74 Shinichi Mochizuki induced by the augmentation   : R   R  . Also, we observe that any choice of representatives in S  of the Δ X  /Δ X  = Gal(X  /X  )-orbits of S  [where we note that the set of such orbits may be naturally identified with S  ] determines an R  -basis of R  [S  ], hence [by considering “Hall bases” — cf., e.g., [Bour], Chapter II, §2.11] an R  -basis of Lie cR  (R  [S  ]). Note that since the natu- ral action of Gal(X  /X  ) on Lie cR  (R  [S  ]) is compatible with the Lie algebra structure, it follows that: This natural action of Gal(X  /X  ) on Lie cR  (R  [S  ]) is given by composing the R  -module structure action Gal(X  /X  ) → R  with the morphism c· : Gal(X  /X  ) → Gal(X  /X  ) given by multiplica- tion by c. In particular, this natural action of Gal(X  /X  ) on Lie cR  (R  [S  ]) factors through the quotient Gal(X  /X  )  c·Gal(X  /X  ) and hence determines on Lie cR  (R  [S  ]) a structure of “induced” c · Gal(X  /X  )-module [in the terminology of the co- homology theory of finite groups]. Thus, we obtain natural, Π X -equivariant isomorphisms ∼   R  [S  ] → R  [S  ] Gal(X /X ) ∼     Lie cR  (R  [S  ]) ⊗ R  c ,  c R  → Lie cR  (R  [S  ]) Gal(X /X ) = Lie cR  (R  [S  ]) c·Gal(X /X ) [where we use superscripts to denote the submodules of invariants with re- spect to the action of the superscripted group]. Moreover, we observe that relative to these natural isomorphisms, the restriction of the natural surjection Lie cR  (R  [S  ])  Lie cR  (R  [S  ]) ⊗ R  c ,  c R  to the submodule of Gal(X  /X  )- invariants induces the endomorphism of the module Lie cR  (R  [S  ]) ⊗ R  c ,  c R  given by multiplication by the order of c · Gal(X  /X  ). Now let us write: def c   New tor S  /S  (2c/2c + 1) = Lie R  (R [S ] ⊗ M X ) ⊗ (Q l /Z l ) (l) def tor   R New tor S  /S  (2c/2c + 1) = New S  /S  (2c/2c + 1) ⊗ R  c , c [where c ≥ 1 is an integer]. Then in light of the above observations [together with Propositions A.1, (iv); 3.1, (iv)], we conclude the following: (A) The natural surjection of Π X -modules tor New tor S  (b + 1/b + 2)  New S  (b + 1/b + 2) admits a factorization tor tor New tor S  (b + 1/b + 2)  New S  /S  (b + 1/b + 2)  New S  /S  (b + 1/b + 2)  New tor S  (b + 1/b + 2) 75 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves [via morphisms of Π X -modules]. Moreover, the natural action of Δ X  on the   module New tor S  /S  (b+1/b+2) factors through the quotient Δ X   Gal(X /X )  c · Gal(X  /X  ) and determines on New tor S  /S  (b + 1/b + 2) a structure of induced c · Gal(X  /X  )-module. (B) The induced morphism on Δ X  -invariants Δ X  Δ X  → New tor = New tor New tor S  (b + 1/b + 2) S  (b + 1/b + 2) S  (b + 1/b + 2) of the [first] natural surjection of (A) factors, in a Π X -equivariant fashion, through the endomorphism tor New tor S  /S  (b + 1/b + 2) → New S  /S  (b + 1/b + 2) tor [hence also through the endomorphism New tor S  (b+1/b+2) → New S  (b+1/b+2)] given by multiplication by the order of c · Gal(X  /X  ). Also, before proceeding, we make the following elementary observation concerning the group cohomology of induced modules:  ab  (C) Suppose that H  = l n · Δ ab X  ⊆ Δ X  , where n is a positive integer. For M a finitely generated Z l -module [which we regard as equipped with the trivial Δ X  -action], write: def (l) H X  = H 1 (Δ X  , M ⊗ M X ) def ∼ H X  = H 1 (Δ X  , M ⊗ M X ) → H 1 (Δ X  , M [Gal(X  /X  )] ⊗ M X ) (l) (l) Then the “trace map” Tr H : H X  → H X  — i.e., the map induced by the morphism of coefficients M [Gal(X  /X  )]  M that maps each element of Gal(X  /X  ) to 1 — factors through the endomor-  phism of H X  given by multiplication by l n [cf. Remark 39 below]. [Indeed, to verify (C), we recall that this trace map Tr H is well-known to be dual [via Poincaré duality — cf., e.g., [FK], pp. 135-136] to the pull- back morphism; we thus conclude that, relative to the natural isomorphisms ∼ ∼ ab H X  → Δ ab X  ⊗ M , H X  → Δ X  ⊗ M [arising from Poincaré duality — cf., e.g., Proposition 1.3, (ii)], the trace map corresponds to the natural morphism ab H X  = Δ ab X  ⊗ M → Δ X  ⊗ M = H X  76 Shinichi Mochizuki induced by the inclusion Δ X  ⊆ Δ X  — hence, by the definition of H  , fac-  tors through the endomorphism of H X  given by multiplication by l n . This completes the proof of (C).] Next, let us suppose that we have been given morphisms of (S, ∅, Σ)- admissible coverings of X X  → X ∗ ∗ → X  → X ∗  ∗  ∗ → X   ∗ and write U S   ⊆ X  , U S ∗ ⊆ X , U S  ∗ ⊆ X for the open subscheme ∗ ∗ determined, respectively, by the complements of the sets S  , S , S of closed ∗ ∗  points of X , X , X that lie over points of S. Also, let us assume that the open subgroups Δ X  ⊆ Δ X  , Δ X ∗ ⊆ Δ X  , Δ X  ⊆ Δ X  , Δ X  ∗ ⊆ Δ X  arise, respectively, from open subgroups ⊆ H H  = l n · Δ ab X  ∗  ∗  ∗ H  = l n · Δ ab X  ⊆ H ∗ def ∗ ∗ = l n ∗ = l n · Δ ab ⊆ Δ ab X ∗ X ∗ ab · Δ ab X  ⊆ Δ X  def — where n = n  − c, n = n  − c; we suppose that n  > 2c, n  > c are “sufficiently large” positive integers, to be chosen below. Then we wish to apply the theory developed above [in particular, the observations (A), (B), (C)] by taking “X  → X  ” in this theory to be various subcoverings of X  → X  . Now let us compute the cohomology of Π X via the Leray spectral sequence associated to the surjection Π X  Π X /Δ X  ∗ . Suppose that c has been chosen  ∗ so that b + 1 = 2c. Then by applying (A) to the covering “X  → X ”  ∗ (respectively, “X  → X ”), we conclude that Δ X ∗ (respectively, Δ X ∗ ) tor acts trivially on New tor S  /S  ∗ (b + 1/b + 2) (respectively, New S  /S ∗ (b + 1/b + 2)). Also, it follows immediately from the definitions that we have a natural Π X - tor equivariant surjection New tor S  /S  ∗ (b + 1/b + 2)  New S  /S ∗ (b + 1/b + 2). ∗ Now, by applying (A) to the covering “X  → X ” and (C) to the covering ∗  ∗ → X ”, we conclude that the Π X -equivariant natural morphism “X tor 1 ∗ , New  H 1 (Δ X ∗ , New tor S  /S ∗ (b + 1/b + 2)) → H (Δ X S /S  ∗ (b + 1/b + 2)) [which maps the image of η S  to the image of η S  !] factors through a “trace ∗ ∗ → X ”, hence in particular, through the map” as in (C) for the covering “X endomorphism of H 1 (Δ X  ∗ , New tor S  /S ∗ (b+1/b +2)) [a module whose submod- ule of Π X -invariants is finite, by Proposition 3.2, (i)] given by multiplication ∗ by n , in a Π X -equivariant fashion. Thus, by taking n  to be “sufficiently large”, we conclude that the image of η S  in H 1 (Δ X ∗ , New tor S  /S  ∗ (b+1/b +2)) is zero. Now I claim that the image of η S  in H 1 (Δ X  , New tor S  /S  (b + 1/b + 2)) [obtained by applying the surjection tor New tor S  (b + 1/b + 2))  New S  /S  (b + 1/b + 2) Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 77 of (A) applied to the covering “X  → X  ”] is zero. Indeed, note that it follows immediately from the definitions that we have a natural surjection tor New tor S  /S ∗ (b+1/b+2)  New S  /S  (b+1/b+2) [induced, in effect, by the inclu-  ∗ sion Gal(X  /X ) → Gal(X  /X  )]. Thus, since we have already shown that the image of η S  in the cohomology module H 1 (Δ X  ∗ , New tor S  /S ∗ (b + 1/b + 2)) 1 is zero, it follows immediately that the image of η S  in H (Δ X ∗ , New tor S  /S  (b+ 1/b + 2)) is zero, hence that the image in question in the claim arises from a class ∈ H 1 (Gal(X ∗ Δ ∗ /X  ), (New tor ) S  /S  (b + 1/b + 2)) X = H 1 (Gal(X ∗ /X  ), New tor S  /S  (b + 1/b + 2)) = 0 [where the last cohomology module vanishes since, by (A) applied to the cov-  ∗  ering “X  → X  ”, New tor S  /S  (b + 1/b + 2) is an induced Gal(X /X )-module]. This completes the proof of the claim. Thus, in summary, we conclude that the image of η S  in the cohomology module H 1 (Π X , New tor S  /S  (b + 1/b + 2)) arises from a unique class in ∼ Δ X  ) → H 1 (Π X /Δ X  , New tor H 1 (Π X /Δ X  , (New tor S  /S  (b+1/b+2)) S  /S  (b+1/b+2)) which maps to the unique class in H 1 (Π X /Δ X  , New tor S  (b + 1/b + 2)) [a module which is finite, by Proposition 3.2, (i)] that gives rise to η S  via a morphism that factors through the endomorphism given by multiplication by the order of c · Gal(X  /X  ) [cf. (A), (B) applied to the covering “X  → X  ”]. In particular, by taking n  to be “sufficiently large”, we may conclude that η S  = 0, as desired. That is to say: TOR≤b+1 This completes the proof that the two inclusions Π ≤b U S → Π U S TOR≤b+1 differ by conjugation by an element of Ker(Π U  Π LIE≤b+ ). U S S In particular, the proof of assertions (i), (ii) is complete. Next, we consider assertion (iii). First, let us observe that when b = 1, assertion (iii) follows immediately from [the “pro-l version” of the argument applied to prove] Proposition 2.3, (i) [cf. the discussion preceding Proposition 3.4]. Thus, in the remainder of the proof of assertion (iii), we assume that b ≥ 2. ≤b+1  Π ≤b Note that since the elements of Ker(Π U U S ) manifestly commute with S , it follows from conditions (a), (b), the fact that b ≥ 2, the elements of I x ≤b+1 ∗ and the assumption that S = S ∗ is of cardinality one that β induces the identity on Π ≤b+1 [csp] [cf. the proof of assertion (ii) above]. Thus, to complete the proof U S of assertion (iii), it suffices to show that the compatible system of classes λ S  ∈ H 1 (Π X , New S  (b + 1/b + 2)) determined by β [cf. Proposition 3.4, (i); 3.1, (iv)] vanishes. Note that since (l) (Δ X ) ab is of “weight ≤ 1”, and New S  (b + 1/b + 2) is of “weight b + 1 ≥ 3” [cf. 78 Shinichi Mochizuki Proposition 3.2, (i)], it follows immediately from the Leray spectral sequence for Π X  G k that we have a natural isomorphism ∼ H 1 (G k , (New S  (b + 1/b + 2)) Δ X ) → H 1 (Π X , New S  (b + 1/b + 2)) [where the superscript “Δ X ” denotes the Δ X -invariants] and that the module H 1 (G k , (New S  (b + 1/b + 2)) Δ X ) is finite. Thus, to show that the λ S  = 0, it suffices to show that the inverse limit lim (New S  (b + 1/b + 2)) Δ X ←− X  [where X  , S  are as described in the proof of assertions (i), (ii)] is zero. But this follows from observation (B) of the proof of assertions (i), (ii). This completes the proof of assertion (iii). TOR≤b+1 , Next, we consider assertion (iv). In light of the definition of Π U S it suffices to show that any compatible system of D X -invariant [relative to the diagonal action of D X ] classes κ S  ∈ New tor S  (b + 1/b + 2) [where X  , S  are as described in the proof of assertions (i), (ii)] lies in the image of I x ∗ [U S ] ⊗ (Q l /Z l ) if b = 1 and vanishes if b ≥ 2. [Here, we note that since New tor S  (b + 1/b + 2) = 0 when b is even, we may assume without loss of generality that b is odd.] To do this, let X  , X  , S  , S  be as in (A), (B). Now we would like to apply the theory of the Appendix [cf., especially, Theorem A.1] to the present situation. To do this, it is necessary to specify the data “(i), (ii), (iii), (vi), (vii), (viii), (ix), (x), (xi), (xii)” [cf. the discussion of the Appendix] to which this theory is to be applied. We take the “d” of Theorem A.1 to be such that 2d = b+1 [so the fact that b is odd implies that d ≥ 2 whenever b ≥ 2] and the prime number “l” of “(i)” to be the prime number l of the present discussion. We take the profinite group “Δ” of “(ii)” to be the quotient of the group Δ X by the kernel of the quotient ab (Δ X ⊇) Δ X   Δ ab X   Δ X  ⊗Z l ; this group “Δ” surjects onto Δ X /Δ X  , which we take to be the quotient group “G” of “(ii)”, with kernel Δ ab X  ⊗ Z l , which we take to be the subgroup “V ” of “(ii)”. Here, we recall that the condition of “(ii), (c)” concerning the regular representation follows immediately from [Milne], p. 187, Corollary 2.8 [cf. also [Milne], p. 187, Remark 2.9], in light of our assumption that X is proper hyperbolic, hence of genus ≥ 2. We take the profinite group “Γ” of “(ix)” to be the image G k  ⊆ G k of Π X  in G k [so “Γ” acts naturally on “Δ”, “G”, “H”]. Thus, “Δ Γ ” may be thought of as a quotient of Π X × G k G k  , hence also as a quotient of D X × G k G k  . Note that by consideration of “weights”, it follows that G k  (New tor S  (b + 1/b + 2)) is finite, hence annihilated by some finite power of l, which we take to be the number “N ” of “(iii)”. We take the covering X  → X  of (A), (B) to 79 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves be any (S, ∅, Σ)-admissible covering such that the resulting covering X k  → X k  is the covering determined by the resulting subgroup “l n · V ⊆ Δ” [cf. the statement of Theorem A.1], so “J ” may be identified with Gal(X  /X  ). Next, we take the “G-torsor E G ” of “(vi)” to be S  and the “H-torsor E H ” of “(vii)” to be S  ; thus, the natural surjection S   S  determines a surjection “E H  E G ” as in “(viii)”. Note that S  (respectively, S  ) may be thought of as a Δ U S   -orbit (respectively, Δ U S   -orbit) [via the action by conjugation] of the conjugacy class of subgroups of Δ U S determined by I x ∗ [U S ] ⊆ Δ U S . In particular, it follows that the particular member of this conjugacy class constituted by the subgroup I x ∗ [U S ] ⊆ Δ U S determines a particular element e H ∈ E H (respectively, e G ∈ E G ) as in “(xi)”. Moreover, the diagonal action of D X — hence also of D X × G k G k  ⊆ D X — on Δ U S determines an action of D X × G k G k  ⊆ D X on E H , E G that fixes e H , e G , and [as is easily verified] factors through the quotient “Δ Γ ” of D X × G k G k   Π X × G k G k  ; in particular, we obtain continuous actions of “Δ Γ ” on “E G ”, “E H ” as in “(x)”. Finally, we (l) take the “Γ-module Λ” of “(xii)” to be the d-th tensor power of M X ⊗ (Q l /Z l ). This completes the specification of the data necessary to apply Theorem A.1. Thus, by applying Theorem A.1 to the composite of the second and third surjections in the factorization of (A), we conclude that since κ S  is D X - invariant, it follows that κ S  ∈ New tor S  (b + 1/b + 2) G k  maps to an element [i.e., κ S  ] of N · New tor = 0 when b ≥ 2 and S  (b + 1/b + 2)  to an element [i.e., κ S ] in the image of I x ∗ [U S ] ⊗ (Q l /Z l ) when b = 1. This completes the proof of assertion (iv). Finally, we consider assertion (v). It is immediate from the definitions that the various quotients in question are cuspidally pro-l. That these quotients are the maximal cuspidally pro-l quotients follows from the construction of Δ ≤∞ U S (l) (l) and the easily verified fact that each Δ U  injects into Lin(Δ U  (1/∞))(Q l ). S  S  Finally, the asserted slimness follows from the fact that the profinite groups in question may be written as inverse limits of profinite groups that admit (l) (l) normal open subgroups [with trivial centralizers] — namely, “Δ U  ”, “(Π U  ) † ”, S  S  “Δ U X  ×X  ”, “(Π U X  ×X  ) † ” — which are slim, by Proposition 1.6, (i), (iii) [which (l) (l) implies that the quotients Δ U X  ×X   Δ X  , (Π U X  ×X  ) †  (Π X  ) † , as well as the kernels of these quotients, are slim]. (l) (l) (l) (l) Remark 38. Proposition 3.6, (iii), may be regarded as a “higher order, pro-l analogue” of Proposition 2.3, (i). Remark 39. It is important to note that if one omits [as was, mis- takenly, done in an earlier version of this paper] the hypothesis that S 0 = ∅, 80 Shinichi Mochizuki then it no longer holds that the image of the trace map “Tr H : H X  → H X  ” [appearing in the proof of Proposition 3.6, (i), (ii)] lies in l n · H X  . Indeed, this phenomenon may be understood by considering the trace map on first étale cohomology modules with Z l -coefficients associated to the l n -th power map G m → G m on the multiplicative group G m over k — a map which, as an easy computation reveals, is surjective. We are now ready to prove the main technical result of the present §3: Theorem 3.1. (Reconstruction of Maximal Cuspidally Pro-l Extensions) Let X, Y be proper hyperbolic curves over a finite field; denote the base fields of X, Y by k X , k Y , respectively. Suppose further that we def def have been given points x ∗ ∈ X(k X ), y ∗ ∈ Y (k Y ); write S = {x ∗ }, T = {y ∗ } def def U S = X\S, V T = Y \T . Let Σ be a set of prime numbers that contains at least one prime number that is invertible in k X , k Y ; thus, Σ determines various quotients Π U S , Π X , Π U X×X , Π X×X , Π V T , Π Y , Π U Y ×Y , Π Y ×Y [cf. Proposition 1.6, (iii); the discussion preceding Proposition 1.5] of the étale fundamental groups of U S , X, U X×X , X × X, V T , Y , U Y ×Y , Y × Y , re- spectively. Also, we write Π X  G k X , Π Y  G k Y for the quotients determined by the respective absolute Galois groups of k X , k Y . Let ∼ α : Π X → Π Y be a Frobenius-preserving [hence also quasi-point-theoretic — cf. Re- mark 10] isomorphism of profinite groups that maps the decomposition group of x ∗ in Π X [which is well-defined up to conjugation] to the decomposition group of y ∗ in Π Y [which is well-defined up to conjugation]. Then for each prime l ∈ Σ such that l = p, there exist commutative diagrams Π ≤∞ U S ∞ −→ Π ≤∞ V T α ⏐ ⏐ Π X α −→ Π ≤∞ U X×X ⏐ ⏐ ⏐ ⏐ Π Y Π X×X α × ∞ −→ Π ≤∞ U Y ×Y ⏐ ⏐ α×α −→ Π Y ×Y ≤∞ ≤∞ ≤∞ — in which Π U S  Π ≤∞ U S , Π U X×X  Π U X×X , Π V T  Π V T , Π U Y ×Y  Π U Y ×Y are the maximal cuspidally pro-l quotients [cf. Proposition 3.6, (v)]; Π X×X ∼ = Π X × G kX Π X , Π Y ×Y ∼ = Π Y × G kY Π Y ; the vertical arrows are the × natural surjections; α ∞ , α ∞ are isomorphisms, well-defined up to composi- tion with a cuspidally inner automorphism, that are compatible, relative to the natural surjections c-ab,l Π ≤∞ U S  Π U S ; c-ab,l Π ≤∞ U X×X  Π U X×X ; c-ab,l Π ≤∞ ; V T  Π V T c-ab,l Π ≤∞ U Y ×Y  Π U Y ×Y 81 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves — where we use the superscript “c-ab, l” to denote the respective maximal cuspidally pro-l abelian quotients — with the isomorphisms ∼ c-ab Π c-ab U S → Π V T ; ∼ c-ab Π c-ab U X×X → Π U Y ×Y of Theorem 2.1, (i); Theorem 1.1, (iii), respectively. Moreover, α ∞ (respec- tively, α × ∞ ) is compatible, up cuspidally inner automorphisms, with the de- ≤∞ composition groups of x ∗ , y ∗ in Π ≤∞ U S , Π V T (respectively, with the images of the ≤∞ decomposition groups D X , D Y in Π U X×X , Π ≤∞ U Y ×Y ). Finally, this condition of “compatibility with decomposition groups”, together with the condition of mak- ing the above diagrams commute, uniquely determine the isomorphisms α ∞ , × α × ∞ , up to composition with a cuspidally inner automorphism; in particular, α ∞ is compatible, up to composition with a cuspidally inner automorphism, with ≤∞ the automorphisms of Π ≤∞ U X×X , Π U Y ×Y given by switching the two factors. Proof. First, let us consider the isomorphism [i.e., more precisely: a spe- cific member of the cuspidally inner equivalence class of isomorphisms] ∼ c-ab,l α c-ab,l : Π c-ab,l U X×X → Π U Y ×Y ∼ c-ab arising from the isomorphism Π c-ab U X×X → Π U Y ×Y of Theorem 1.1, (iii). Recall that since α is Frobenius-preserving, it is quasi-point-theoretic [cf. Remark 10], and that α c-ab,l is compatible with the images of D X , D Y , which we denote by (l) (l) D X , D Y . Thus, we may assume without loss of generality that our choices of decomposition groups D x ∗ [U S ] ⊆ Π U S , D y ∗ [V T ] ⊆ Π V T , as well as our choices c-ab,l of splittings G k X → D x ∗ [U S ], G k Y → D y ∗ [V T ], have images in Π c-ab,l U X×X , Π U Y ×Y that correspond via α c-ab,l . In particular, it follows that α c-ab,l maps Π c-ab,l ⊆ U S c-ab,l c-ab,l Π c-ab,l isomorphically onto Π ⊆ Π . U X×X V T U Y ×Y In the following argument, let us identify the “Lin U S (1/∞)”, “Lin X (1/∞)” portions of Δ Lie U S with the [completions, relative to the natural filtration topol- ogy, of the] corresponding graded objects “Gr Q l (−)(1/∞)” via the Galois in- variant splittings of Proposition 3.2, (ii), and similarly for V T . Then, in light of our assumption that α is Frobenius-preserving, it follows immediately from the naturality of our constructions [cf., especially, Proposition 3.2, (iii)] that α induces, for each Z b ≥ 1, compatible isomorphisms ∼ LIE α LIE : Π LIE U S → Π V T ; ∼ α LIE≤b : Π LIE≤b → Π LIE≤b U S V T which are, moreover, compatible [with respect to the natural projections to Π X , Π Y ] with the isomorphism α. Moreover, it follows from the construction of “Π LIE≤b ” that the latter displayed isomorphism maps D x LIE≤b ⊆ Π LIE≤b bijec- U S (−) ∗ ∼ ⊆ Π LIE≤b , and that the resulting isomorphism D x LIE≤b → D y LIE≤b tively onto D y LIE≤b V T ∗ ∗ ∗ induces an isomorphism ∼ → D y ≤b D x ≤b ∗ ∗ 82 Shinichi Mochizuki which is compatible [again by construction!] with the respective Frobenius elements “F k ” on either side. Next, let us observe that since the isomorphism α c-ab,l induces an isomor- ∼ → Π c-ab,l that is compatible with the images of the decomposi- phism Π c-ab,l U S V T tions groups D x ∗ [U S ], D y ∗ [V T ] and Frobenius elements in these decomposition groups, it follows immediately that for corresponding [i.e., via α] (S, ∅, Σ)- , (T, ∅, Σ)-admissible coverings X  → X, Y  → Y [which induce coverings Lie≤2 ∼ U S   → U S , V T  → V T ], α c-ab,l induces an isomorphism Δ U → Δ V Lie≤2  T  S  which is compatible with α LIE≤2 . Moreover, although Δ Lie≤2 , Δ V Lie≤2 are U  T  S  Lie≤2  H X , Δ V Lie≤2  H Y are eas- not center-free, the semi-direct products Δ U  T  S  ily seen to be center-free [cf. Proposition 1.6, (i)], for arbitrary open sub- groups H X ⊆ G † k X , H Y ⊆ G † k Y [where the daggers are as in Proposition 1.6, (respectively, Π LIE≤2 ) is an inverse (i)] that correspond via α. Since Π LIE≤2 U S V T limit of topological groups that admit normal closed subgroups of the form Lie≤2 Δ U  H X (respectively, Δ V Lie≤2  H Y ), we thus conclude [by applying the  T  S  extension “1 → (−) → Aut(−) → Out(−) → 1” of §0 to these normal closed ∼ subgroups] that the isomorphism Π c-ab,l → Π c-ab,l induced by α c-ab,l is com- U S V T patible — relative to the natural inclusions ∼ LIE≤2 → Π ≤2 ; Π c-ab,l U S U S → Π U S ∼ LIE≤2 Π c-ab,l → Π ≤2 V T V T → Π V T ∼ [cf. the discussion preceding Proposition 3.4] — with α LIE≤2 : Π LIE≤2 → Π LIE≤2 . U S V T In fact, since 3 is odd, it follows immediately from the definitions that the modules “New Q S  (3/4)” vanish, hence [cf. Definition 3.3, (ii)] that we have ∼ → Π LIE≤2+ , which implies [cf. Proposition 3.4, (i)] an isomorphism Π LIE≤3 U S U S ≤3 ∼ that we have an isomorphism Π U S → Π ≤2 U S [and similarly for V T ]. Thus, by ∼ Proposition 3.4, (iii), it follows that the isomorphism Π c-ab,l → Π c-ab,l induced U S V T c-ab,l is compatible — relative to the natural inclusions by α ∼ LIE≤3 → Π ≤3 ; Π c-ab,l U S U S → Π U S ∼ LIE≤3 Π c-ab,l → Π ≤3 V T V T → Π V T ∼ — with α LIE≤3 : Π LIE≤3 → Π LIE≤3 . U S V T LIE Next, let us observe that the diagonal actions of D X , D Y on Π LIE U S , Π V T (l) (l) (l) clearly factor through D X , D Y [hence determine “diagonal actions” of D X , (l) LIE D Y on Π LIE U S , Π V T ]. Moreover, by what we have already shown concerning the (l) compatibility of α LIE≤3 with α c-ab,l [cf. also the compatibility of α c-ab,l with D X , (l) D Y ] and the compatibility of α c-ab,l with the decomposition groups D x ∗ [U S ], D y ∗ [V T ], it follows [cf. Remarks 36, 37] that the conditions (a), (b), (c) of Proposition 3.5 are compatible with α LIE , hence that α LIE is compatible with (l) (l) LIE the diagonal actions of D X , D Y on Π LIE U S , Π V T [relative to the isomorphism (l) ∼ (l) D X → D Y induced by α c-ab,l ]. ≤b Now I claim that the isomorphism α LIE≤b maps Π ≤b U S bijectively onto Π V T , Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 83 thus inducing a compatible inverse system [parametrized by b] of isomorphisms ∼ ≤b α ≤b : Π ≤b U S → Π V T ≤b that are compatible [with respect to the natural projections Π ≤b U S  Π X , Π V T  Π Y ] with α. To verify this claim, we apply induction on b. The case b = 1 is vacuous; the case b = 2 follows from what we have already shown concerning the compatibility of α LIE≤2 with α c-ab,l . Thus, we assume that b ≥ 2, and that the claim has been verified for “b” that are ≤ the b under consideration. Now observe that by Propositions 3.4, (iii); 3.6, (ii), it follows that the isomorphism ∼ → Π LIE≤b+1 Π LIE≤b+1 U S V T maps Π ≤b+1 bijectively onto a Ker(Π LIE≤b+1  Π LIE≤b+ )-conjugate of Π V ≤b+1 . U S V T V T T  In particular, by conjugating by an appropriate element γ ∈ Ker(Π LIE≤b+1 V T Π V LIE≤b+ ), we obtain an isomorphism T ∼ ≤b+1 → Π V ≤b+1 β b+1 : Π U S T that is compatible with α ≤b and, moreover, [since γ commutes with I y ≤b+1 ] ∗ ≤b+1 maps I x ≤b+1 bijectively onto I . Note that by Propositions 3.4, (i); 3.6, y ∗ ∗  Π ≤b+ (iii), it follows that the choice of γ is unique, modulo Ker(Π V ≤b+1 V T ). In T TOR≤b+1 TOR≤b+1 of γ in Π V T is uniquely determined. particular, the image δ ∈ Π V T On the other hand, since α LIE is compatible with the diagonal actions of (l) (l) LIE D X , D Y on Π LIE U S , Π V T , it follows immediately, by “transport of structure”, (l) that δ is fixed by the diagonal action of D Y . But, by Proposition 3.6, (iv), this implies that δ = 0. This completes the proof of the claim. ∼ ≤∞ Thus, we obtain an isomorphism α ∞ : Π ≤∞ U S → Π V T as in the statement of ≤∞ Theorem 3.1. Next, let us recall that Δ ≤∞ U S , Δ V T are slim [cf. Proposition 3.6, (v)]. Thus, since this isomorphism α ∞ is compatible with the diagonal actions ∼ (l) (l) ≤∞ of D X , D Y , we may apply the isomorphism Aut(Δ ≤∞ U S ) → Aut(Δ V T ) induced by α ∞ to obtain — i.e., by pulling back the extension ≤∞ ≤∞ 1 → Δ ≤∞ U S → Aut(Δ U S ) → Out(Δ U S ) → 1 [cf. §0] via the homomorphism (D X ) Π X → Out(Δ ≤∞ U S ) (l) arising from the diagonal action [and similarly for Δ ≤∞ V T ] — an isomorphism ∼ ≤∞ ≤∞ α × : Π → Π as in the statement of Theorem 3.1. Here, we note ∞ U X×X U Y ×Y × that the “cuspidally inner indeterminacy” of α ∞ , α ∞ that is referred to in the statement of Theorem 3.1 arises from the “cuspidally inner indeterminacy” in the choice of corresponding decomposition groups D x ∗ [U S ], D y ∗ [V T ] [more pre- c-ab,l ≤∞ cisely: the images of these groups in Π ≤∞ U S , Π V T , as opposed to just in Π U S , 84 Shinichi Mochizuki Π c-ab,l ]. Finally, we observe that the asserted uniqueness follows immediately V T by considering eigenspaces relative to the Frobenius actions [cf. Proposition 3.2, (ii)], together with the construction of the isomorphism α LIE [cf. also Propositions 1.10, (i); 2.3, (i)]. Remark 40. The argument of the proof of Theorem 3.1 involving Proposition 3.6, (iv), may be regarded as a sort of “higher order analogue” of the argument applied in the proof of Theorem 1.1, (iii), involving Lemma 1.1; Proposition 1.8, (v). Remark 41. At first glance, it may appear that the portion of The- orem 3.1 concerning α × ∞ may only be concluded when X(k X ), Y (k Y ) are ≤∞ † † nonempty. In fact, however, since (Π ≤∞ U X×X ) , (Π U Y ×Y ) are slim [cf. Proposi- tion 3.6, (v)], it follows that the portion of Theorem 3.1 concerning α × ∞ may be concluded even without assuming that X(k X ), Y (k Y ) are nonempty, by ap- plying Theorem 3.1 after passing to corresponding [via α] finite extensions of k X , k Y [cf. Remark 5]. Remark 42. It seems reasonable to expect that, when, say, Σ = {l}, the techniques applied in the proof of Theorem 3.1, together with the theory of [Mtm], should allow one to reconstruct the [geometrically pro-Σ] étale funda- mental groups of the various configuration spaces [i.e., finite products of copies of X over k X , with the various diagonals removed] “group-theoretically” from Π X [under, say, an appropriate hypothesis of “Frobenius-preservation” as in Theorem 3.1]. This topic, however, lies beyond the scope of the present paper. Remark 43. If the “cuspidalization of configuration spaces” [cf. Re- mark 42] can be achieved, then it seems likely that by applying an appropriate “specialization” operation, it should be possible to generalize Theorem 3.1 to the case where S, T are subsets of arbitrary finite cardinality. Remark 44. One essential portion of the proof of Theorem 3.1 is the Galois invariant splitting of Proposition 3.2, (ii). Although it does not appear likely that such a splitting exists in the case of a nonarchimedean local base field [cf., e.g., the theory of [Mzk4]], it would be interesting to investigate the extent to which a result such as Theorem 3.1 may be generalized to the nonar- chimedean local case, perhaps by making use of some sort of splitting such as the Hodge-Tate decomposition, or a splitting that arises via crystalline methods. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 85 In the context of absolute anabelian geometry over nonarchimedean local fields, however, such p-adic Hodge-theoretic splittings might not be available, since the isomorphism class of the Galois module “C p ” is not preserved by arbitrary automorphisms of the absolute Galois group of a nonarchimedean local field [cf. the theory of [Mzk3]]. The development of the theory underlying Theorem 3.1 was motivated by the following important consequence: Corollary 3.1. (Total Global Green-compatibility) In the sit- uation of Theorem 1.1, (iii) [in the finite field case], suppose further that Σ † = Primes † , and that X, Y are Σ-separated [which implies that α is Frobenius-preserving and point-theoretic — cf. Remarks 9, 10]. Then the isomorphism α is totally globally Green-compatible. Proof. Indeed, we may apply Theorem 3.1 to the isomorphism α of Theo- rem 1.1, (iii), and arbitrary choices of sets of cardinality one S = {x ∗ }, T = {y ∗ } that correspond via α. Let l ∈ Σ † . Then let us observe that the quotient Π U S  Π ≤∞ U S satisfies the following property: If Π U S  Q is a finite quotient of Π U S such that for some quotient Q  Q  whose kernel has order a power of l, Π U S  Q  factors through Π U S  Π ≤∞ U S , then Π U S  Q also factors through Π U S  . Π ≤∞ U S A similar statement holds for the quotient Π V T  Π ≤∞ V T . In light of this obser- vation, together with our assumption that Σ † = Primes † [which implies that α is Frobenius-preserving], it follows that the reasoning of [Tama], Corollary 2.10, Proposition 3.8 [cf. also Remark 10 of the present paper], may be applied to the isomorphism ∼ ≤∞ α ∞ : Π ≤∞ U S → Π V T of Theorem 3.1 to conclude that the isomorphism α ∞ maps the set of de- composition subgroups of the domain bijectively onto the set of decomposition subgroups of the codomain. On the other hand, sorting through the definitions, the datum of the lift- ing of a decomposition group of Π X , Π Y corresponding to a point that does not belong to S, T to a [noncuspidal] decomposition group of the domain or c-ab,l codomain of α ∞ determines, by projection to Π c-ab,l , the l-adic portion U S , Π V T of the Green’s trivialization associated to this point and the unique point of S or T . Since l is an arbitrary element of Σ † = Primes † , and the points x ∗ , y ∗ are arbitrary points that correspond via α, this shows that α is globally Green- compatible. That α is totally globally Green-compatible follows by applying this 86 Shinichi Mochizuki argument to the isomorphism induced by α between open subgroups of Π X , Π Y . Theorem 3.2. (The Grothendieck Conjecture for Proper Hy- perbolic Curves over Finite Fields) Let X, Y be proper hyperbolic curves over a finite field; denote the base fields of X, Y by k X , k Y , re- spectively. Write Π X , Π Y for the étale fundamental groups of X, Y , respectively. Let ∼ α : Π X → Π Y be an isomorphism of profinite groups. Then α arises from a uniquely de- termined commutative diagram of schemes  X ∼ → ⏐ ⏐ ⏐ ⏐ X Y  ∼ → Y in which the horizontal arrows are isomorphisms; the vertical arrows are the pro-finite étale universal coverings determined by the profinite groups Π X , Π Y . Proof. Theorem 3.2 follows formally from Corollaries 2.1, 3.1; Remarks 9, 10; Proposition 2.2, (ii). Appendix. Free Lie Algebras In this Appendix, we discuss various elementary facts concerning free Lie algebras that are necessary in §3. In particular, we develop a sort of “higher order analogue” of the theory developed in Lemma 1.1. Proposition A.1. (Free Lie Algebras) Let R be a commutative ring with unity; V a finitely generated free R-module. Write Lie R (V ) for the free Lie algebra over R associated to V ; for Z b ≥ 1, denote by Lie bR (V ) ⊆ Lie R (V ) the R-submodule generated by the “alternants of degree b” [cf. [Bour], Chapter II, §2.6]. Also, we shall denote by U R (V ) the enveloping algebra of Lie R (V ). [Thus, we have a natural inclusion Lie R (V ) → U R (V ).] Then: (i) Each Lie bR (V ) is a finitely generated free R-module. Moreover, there is Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 87 ∼ a natural isomorphism V → Lie 1 R (V ). (ii) Let v ∈ V be a nonzero element such that the quotient module V /R · v is free. Then the centralizer of v in U R (V ) is equal to the R-submodule of U R (V ) generated by the nonnegative powers of v. In particular, if R is a field of characteristic zero, then the centralizer of v in Lie R (V ) is equal to R · v. (iii) Suppose that the rank of V over R is ≥ 2. Then the Lie algebra Lie R (V ) is center-free. In particular, the adjoint representation of Lie R (V ) is faith- ful. (iv) Let R  be an R-algebra which is finitely generated and free as an R-module. Let φ : R   R be a surjection of R-algebras; suppose that V = V  ⊗ R  ,φ R, for some finitely generated free R  -module V  [so we obtain a natu- ral surjection V   V compatible with φ]. Then the natural surjection V   V induces a surjection of R-modules Lie bR (V  )  Lie bR (V ) that factors as a com- posite of natural surjections as follows: Lie bR (V  )  Lie bR  (V  )  Lie bR (V ) Here, the first arrow of this factorization is the arrow naturally induced by observing that every Lie algebra over R  naturally determines a Lie algebra over R; the second arrow of this factorization is the arrow functorially induced by the natural φ-compatible surjection V   V . Finally, this second arrow ∼ induces an isomorphism Lie bR  (V  ) ⊗ R  ,φ R → Lie bR (V ). Proof. Assertion (i) follows immediately from [Bour], Chapter II, §2.11, Theorem 1, Corollary. Assertion (ii) follows from the well-known structure of the enveloping algebra U R (V ) [i.e., the natural isomorphism of U R (V ) with the free associative algebra determined by V over R; the fact that when R is a field of characteristic zero, the image of Lie R (V ) in U R (V ) may be identified with the set of primitive elements — cf. [Bour], Chapter II, §3, Theorem 1, Corollaries 1,2], by considering the effect on “words” of forming the commutator with v — cf. the argument of [Mtm], Proposition 3.1 [which is given only in the case where R is a field of characteristic zero, but does not, in fact, make use of this assumption on R in an essential way]. Assertion (iii) follows immediately from assertion (ii) [by allowing the element “v” of assertion (ii) to range over the elements of an R-basis of V ]. Assertion (iv) follows formally from the universal property of a free Lie algebra, together with the well-known functoriality of a free Lie algebra with respect to tensor products [cf. [Bour], Chapter II, §2.5, Proposition 3]. Next, let us suppose that we have been given data as follows: 88 Shinichi Mochizuki (i) a prime number l; (ii) a profinite group Δ that admits an normal open subgroup V ⊆ Δ such that the following conditions are satisfied: (a) V is abelian [so we shall regard V as a module]; (b) the topological module V is a finitely generated free R- def module, where we write R = Z l ; (c) the resulting action of the finite group def def G = Δ/V on V determines a G-module V Q l = V ⊗ Q l that contains the regular representation of G; (iii) a positive power N of l; (iv) a collection of [not necessarily distinct!] elements g 1 , . . . , g d ∈ G [where d ≥ 1 is an integer] of G at least one of which is not equal to the identity element. Write def ζ = d  (1 − g i ) ∈ R[G] i=1 [where R[G] is the group ring of G with coefficients in R]. Then we have the following result: Lemma A.1. (Nontriviality of a Certain Operator) There exists an integer n ≥ 1 such that the order |J ζ | of the image J ζ ⊆ J def of the action of ζ on [the finite group] J = V ⊗ (Z/l n Z) is divisible by N . Proof. Indeed, since the G-module V Q l contains the regular representation [cf. condition (ii), (c)], it follows that the image of the action of ζ on V Q l is a nonzero Q l -vector space, hence that the image of the action of ζ on the finitely generated free R-module V [cf. condition (ii), (b)] contains a rank one free R-module. Now Lemma A.1 follows immediately. def Next, let J ζ ⊆ J be as in Lemma A.1; write H = Δ/(l n · V ) [so J ⊆ H, H/J = G]. Also, let us assume that we have been given data as follows: (v) a collection of elements h 1 , . . . , h d ∈ H that lift g 1 , . . . , g d ∈ G; (vi) a G-torsor E G [whose G-action will be written as an action from the left]; 89 Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves (vii) an H-torsor E H [whose H-action will be written as an action from the left]; (viii) a surjection  : E H  E G that is compatible with the natural surjection H  G; (ix) a continuous action of a profinite group Γ on Δ that preserves the def subgroup V ⊆ Δ, hence determines a profinite group Δ Γ = Δ  Γ that acts continuously on G, H [in such a way that the restriction of this action to Δ ⊆ Δ Γ is the action of Δ on G, H by conjugation]; (x) continuous actions of Δ Γ on E G , E H [which will be denoted via super- scripts] that are compatible with the continuous actions of Δ Γ on G, H, as well as with the surjection  and, moreover, induce the trivial action of Γ ⊆ Δ Γ on E G [hence also on G]; (xi) an element [i.e., “basepoint”] e H ∈ E H , whose image via  we denote by e G ∈ E G , such that e H , e G are fixed by the action of Δ Γ on E H , E G . Next, let us write def R J = R[J] for the group ring of J with coefficients in R. Thus, R J is a commutative R- algebra, and we have a natural augmentation homomorphism R J  R [which sends all of the elements of J to 1]. Moreover, if we write def def  M : M H = R[E H ]  M G = R[E G ] for the morphism of R J -modules induced by  on the respective free R-modules with bases given by the elements of E H , E G , then  M induces a natural iso- ∼ morphism M H ⊗ R J R → M G . Thus, it follows from Proposition A.1, (iv), that, for b ≥ 1 an integer, we have [in the notation of Proposition A.1] natural surjections Lie bR (M H )  Lie bR J (M H )  Lie bR (M G ) ∼ the second of which determines a natural isomorphism Lie bR J (M H )⊗ R J R → Lie bR (M G ). Now let P (X 1 , . . . , X d ) be an “alternant monomial of degree d” [i.e., a monomial element of Lie d Z (−) of the free Z-module on the indeterminate symbols X 1 , . . . , X d ] in which each 90 Shinichi Mochizuki X i [for i = 1, . . . , d] appears precisely once. Then P (X 1 , . . . , X d ) determines an element P (g 1 · e G , . . . , g i · e G , . . . , g d · e G ) of Lie dR (M G ). Moreover, by allowing such P (X 1 , . . . , X d ) and g 1 , . . . , g d to vary appropriately, we obtain a Hall basis [cf., e.g., [Bour], Chapter II, §2.11] of Lie dR (M G ) [at least if d ≥ 2; if d = 1, then one must also allow for the unique g 1 to be the identity element]. Similarly, by allowing such P (X 1 , . . . , X d ) and h 1 , . . . , h d ∈ H to vary appropriately, we obtain a Hall basis [again, strictly speaking, if d ≥ 2] of Lie dR J (M H ) of elements of the form P (h 1 ·e H , . . . , h d ·e H ). Lemma A.2. (Relation of Superscript and Left Actions) For any v ∈ V ⊆ Δ ⊆ Δ Γ that maps to j ∈ J, we have P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) v = ζ(j) · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) in Lie dR J (M H ). Proof. Indeed, we compute: P (h 1 · e H , . . . ,h i · e H , . . . , h d · e H ) v = P (h v 1 · e H , . . . , h vi · e H , . . . , h vd · e H ) −1 −1 v v = P (h v 1 · h −1 1 · h 1 · e H , . . . , h i · h i · h i · e H , . . . , h d · h d · h d · e H ) d    [j, h i ] · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) = i=1 = ζ(j) · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) [where we apply the R J -module structure of E H and the fact that e vH = e H [cf. (xi)]]. Next, let us assume that we have also been given the following data: (xii) a topological R-module Λ equipped with a continuous action by Γ [which thus determines, via the natural surjection Δ Γ  Γ, a continuous action by Δ Γ on Λ]. Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 91 Write: def V Γ = V  Γ ⊆ Δ Γ ; def F J = J · P (h 1 · e H , . . . , h d · e H ) ⊆ Lie dR J (M H ); def R[F J ] = R · F J = R J · P (h 1 · e H , . . . , h d · e H ) ⊆ Lie dR J (M H ); def Λ[F J ] = R[F J ] ⊗ R Λ ⊆ Lie dR J (M H )⊗ R ; Λ def F = P (g 1 · e G , . . . , g d · e G ) ∈ Lie dR (M G ); def R[F ] = R · F ⊆ Lie dR (M G ); def Λ[F ] = R[F ] ⊗ R Λ ⊆ Lie dR (M G ) ⊗ R Λ Thus, the natural surjection Lie dR J (M H )  Lie dR (M G ) determines [compatible] natural surjections F J  {F }, R[F J ]  R[F ], Λ[F J ]  Λ[F ]. Also, we observe [cf. the fact that Lie dR J (M H ) is a finitely generated free R J -module] that F J is a J-torsor [relative to the action from the left], hence, in particular, a finite set. Now observe that since V Γ acts trivially on G, e H [cf. (ix), (x), (xi)], it follows immediately that V Γ acts compatibly on F J , R[F J ], Λ[F J ], F , R[F ], Λ[F ], and that the natural action of V Γ on R[G] preserves ζ. In particular, it follows that V Γ preserves J ζ ⊆ J, hence that V Γ acts naturally on the set of orbits (F J ) F ζ of F J with respect to the action of J ζ ; moreover, by Lemma A.2, it follows that this action of V Γ on F ζ factors through the quotient V Γ  Γ. Now let us consider invariants with respect to the various superscript ac- tions under consideration. Let us write Invar(−, −) for the set of invariants of the second argument in parentheses with respect to the superscript action of the group given by the first argument in parentheses. Then any element η ∈ Invar(V Γ , Λ[F J ]) may be regarded as a Λ-valued function on the set F J which descends [cf. Lemma A.2] to a Γ-invariant Λ-valued function on F ζ , i.e., an element η ζ ∈ Invar(Γ, Λ[F ζ ]). Next, let us observe that [since η ζ is Γ-invariant] the sum of the values  ∈ Λ of the Λ-valued function on F ζ determined by η ζ is a Γ-invariant element η ζ ∈ Invar(Γ, Λ). Thus, the sum  η ∈ Λ 92 Shinichi Mochizuki of the values ∈ Λ of the Λ-valued function on F J determined by η satisfies the relation   η = |J ζ | · η ζ  in Λ. But the image of η in Λ[F ] is precisely the element ( η) · F . Thus, since, by Lemma A.1, |J ζ | is divisible by N , we conclude the following: Lemma A.3. image (Monomial-wise Computation of Invariants) The Im(Invar(V Γ , Λ[F J ])) ⊆ Λ[F ] of Invar(V Γ , Λ[F J ]) ⊆ Λ[F J ] in Λ[F ] lies in N · Invar(Γ, Λ[F ]). Thus, by allowing P (X 1 , . . . , X d ) and h 1 , . . . , h d ∈ H as in the above discussion to vary appropriately so as to obtain a Hall basis [again, strictly speaking, if d ≥ 2] of Lie dR J (M H ) of elements of the form P (h 1 ·e H , . . . , h d ·e H ), we conclude the following: Theorem A.1. (Invariants of Free Lie Algebras) Let d ≥ 1 be an integer. Suppose that we have been given data as in (i), (ii), (iii) above. Let n ≥ 1 be an integer that satisfies the property of Lemma A.1 for all [of the finitely many] possible choices of data as in (iv) [relative to the given integer def def def d ≥ 1]; J = V /(l n · V ) ⊆ H = Δ/(l n · V ); R J = R[J]. Suppose that have also been given data as in (vi), (vii), (viii), (ix), (x), (xi), (xii) above; def def def let M H = R[E H ], M G = R[E G ], V Γ = V  Γ (⊆ Δ Γ ). Then the natural surjection Lie dR J (M H ) ⊗ R Λ  Lie dR (M G ) ⊗ R Λ maps Invar(V Γ , Lie dR J (M H ) ⊗ R Λ) into N · Invar(V Γ , Lie dR (M G ) ⊗ R Λ) if d ≥ 2. In a similar vein, the natural surjection M H ⊗ R Λ  M G ⊗ R Λ maps Invar(V Γ , M H ⊗ R Λ) into N ·Invar(V Γ , M G ⊗ R Λ)+Invar(V Γ , Λ)·e G ⊆ M G ⊗ R Λ. References [Bour] N. Bourbaki, Lie Groups and Lie Algebras, Springer Verlag (1989). Absolute Anabelian Cuspidalizations of Proper Hyperbolic Curves 93 [DM] P. 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